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[I. V. Solovyev](https://orcid.org/0000-0002-2010-9877)

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[Hidden ferromagnetism of centrosymmetric antiferromagnets](https://mdr.nims.go.jp/datasets/730f158c-bb7d-48b9-98ca-d6d6887be849)

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PHYSICAL REVIEW B 114, 154413 (2026)Editors’ SuggestionHidden ferromagnetism of centrosymmetric antiferromagnetsI. V. Solovyev *Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS),1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan(Received 11 September 2025; revised 28 July 2026; accepted 13 August 2026; published 15 September 2026)The time-reversal symmetry (T ) breaking is a signature of ferromagnetism, giving rise to such phenomenaas the anomalous Hall effect (AHE) and orbital magnetism. Nevertheless, T can also be broken in certainclasses of antiferromagnets, such as weak ferromagnets or altermagnets, which remain invariant under spatialinversion. In light of this similarity with ferromagnetism, it is tempting to ask whether such an unconventionalantiferromagnetic (AFM) state can be represented as the simplest ferromagnetic one, i.e., within the minimal unitcell containing only one magnetic site. We show that such a representation is possible due to a special form of thespin-orbit (SO) interaction in an antipolar lattice hosting this AFM state. The inversion symmetry constrains theform of the SO interaction, which becomes invariant under the symmetry operation {S|t}, combining the 180◦rotation of spins (S) with the lattice shift t, connecting two antiferromagnetically coupled sublattices. This is thefundamental symmetry property of centrosymmetric antiferromagnets, which justifies the use of the generalizedBloch theorem and transformation to the local coordinate frame with one magnetic site per cell. It naturallyexplains the emergence of AHE and net orbital magnetization, and provides transparent expressions for theseproperties in terms of the electron hoppings and SO interaction operating between AFM sublattices, as well asthe orthorhombic strain, controlling the piezomagnetic response. The idea is illustrated on a number of examplesincluding a two-dimensional square lattice, monoclinic VF4 and CuF2, and RuO2-type materials with the rutilestructure, using for these purposes realistic models derived from first-principles calculations.DOI: 10.1103/myy9-7pm5I. INTRODUCTIONCanonically, the time-reversal symmetry (T ) breaking wassynonymous with ferromagnetism. The magnetic unit cell ofa regular ferromagnet coincides with the chemical one, butthe electronic structure for the states with spins “up” and“down” is different, so that the ferromagnetic (FM) systemis characterized by a finite spin magnetic moment, which hasthe same direction at all magnetic sites. On the contrary, theregular antiferromagnet is characterized by the doubling ofthe unit cell, so that two magnetic sublattices, which in theFM state would be connected by a primitive translation t,are occupied by atoms with opposite directions of spins. Thebasic symmetry of the regular antiferromagnetic (AFM) stateis {T |t}, where T is combined with the lattice translation t ofthe chemical cell. Therefore, although T is microscopicallybroken, it is preserved macroscopically, after averaging overtwo AFM sublattices [1]. The electronic states of such antifer-romagnets are Kramers degenerate.Currently, a great deal of attention is paid to systemsin which the AFM alignment of spins coexists with somefeatures, which are more characteristic for ferromagnets, in-cluding the macroscopic time-reversal symmetry breaking*Contact author: solovyev.igor@nims.go.jpPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.and splitting of AFM bands. Such systems are now called“altermagnets,” to emphasize the distinct character of thesematerials and their principal difference from the conventionalferromagnets and antiferromagnets [2–4].On the other hand, the AFM materials with broken time-reversal symmetry have been known for decades. In particular,in the 1950s, using phenomenological symmetry arguments,Dzyaloshinskii pointed out that there are special types ofantiferromagnets, which can host the phenomena of weak fer-romagnetism [5], piezomagnetism [6], and magnetoelectricity[7]. Contrary to regular antiferromagnetism, the magnetic unitcell of these materials coincides with the chemical one, be-ing the necessary precondition for macroscopic time-reversalsymmetry breaking [1]. A very detailed phenomenologicalclassification has been given by Turov, who suggested divid-ing unconventional antiferromagnets into two major classes,calling them “centrosymmetric” and “centroantisymmetric,”depending on whether the spatial inversion I enters themagnetic group alone or in combination with T [8]. The cen-trosymmetric antiferromagnetism in Turov’s classification,encompassing such phenomena as weak ferromagnetism andpiezomagnetism, has a clear resemblance to what is nowcalled altermagnetism, where the AFM alignment of spinscan induce a spontaneous magnetization and other ferromag-netic properties. The centroantisymmetric antiferromagnetismprovides a general framework for understanding magneto-electricity, which excludes any ferromagnetic behavior in theground state unless an external magnetic or electric fieldis applied. Besides weak ferromagnetism, piezomagnetism,and magnetoelectricity, Turov considers a wide spectrum of2469-9950/2026/114(15)/154413(18) 154413-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-2010-9877https://ror.org/026v1ze26https://crossmark.crossref.org/dialog/?doi=10.1103/myy9-7pm5&domain=pdf&date_stamp=2026-09-15https://doi.org/10.1103/myy9-7pm5https://creativecommons.org/licenses/by/4.0/I. V. SOLOVYEV PHYSICAL REVIEW B 114, 154413 (2026)phenomena expected in the unconventional antiferromagnets.Particularly, in 1962, Turov and Shavrov predicted the anoma-lous Hall effect (AHE) in centrosymmetric antiferromagnets.Moreover, they explicitly stated that AHE is not a side effectof the weak ferromagnetism. Rather, it can be related directlyto the AFM order parameter [8,9]. The Landau theory of al-termagnetism, which was recently formulated in Ref. [10] andapplied to rutile and hexagonal (MnTe) materials, success-fully reproduces the behavior of net magnetization and AHEpredicted in Ref. [8]. Similar symmetry-based arguments canbe found in a recent review article [11]. Turov also pointedout that since magnetoelectricity and weak ferromagnetismbelong to two different classes, they are mutually exclusiveand do not coexist unless they develop in different magneticsublattices [12]. The typical example of such coexistence isGdFeO3, where the Gd sublattice is magnetoelectric, whilethe Fe sublattice is weakly ferromagnetic [13,14].Although many properties of centrosymmetric antiferro-magnets were anticipated at the phenomenological level,recent breakthroughs in this field are related to microscopicunderstanding of these properties, which became largelypossible due to the development of first-principles elec-tronic structure calculations. The main attention is paid tothe search for new materials with large spin splitting ofAFM bands [15–24], which is regarded as a hallmark ofaltermagnetism [2–4,25].Such splitting is a consequence of the rotational symmetryof crystals with several atoms in the unit cell. For nonsym-morphic symmetries, the atomic positions are generated bytwo- or fourfold rotations C, which are combined with latticeshift t connecting different sublattices: {C|t}. The symmorphicanalogs, containing fourfold rotations alone, are also possible,but less common [2,25]. Then, the AFM order is obtained bycombining some of the symmetry operations {C|t} with T , sothat the magnetic unit cell remains the same as the chemicalone, being in line with Dzyaloshinskii’s conjecture [1]. Thenew aspect of the problem, which was overlooked in theearlier stages, is that the {T C|t} symmetry gives rise to spinsplitting of bands in reciprocal space [2,3,25,26]. Today, thissplitting is believed to be the sole signature of ferromagnetismin otherwise AFM materials, which is responsible for AHEand other ferromagnetic phenomena.However, there is another important symmetry, I, whichwas highlighted in Turov’s definition of “centrosymmetricantiferromagnetism,” but somewhat overshadowed by othersymmetry properties in the modern developments of alter-magnetism. Then, which symmetry is more important, forinstance, for the emergence of AHE in AFM substances:{T C|t} or I?In our recent work, dealing with the minimal one-orbitalmodel, where the spin-orbit (SO) interaction replicates theform of Dzyaloshinskii-Moriya (DM) interactions in thenoncentrosymmetric bonds [5,27], we have argued that thealtermagnetic splitting of bands does not play a key role inAHE and orbital magnetism [28]. Namely, T can be brokeneven when the AFM bands are spin-degenerate. Similar be-havior has been found in a more sophisticated multiorbitalmodel for AFM κ-type organic conductors and orthorhom-bically distorted perovskites [19,21]. Furthermore, we haveargued that the fundamental symmetry of such an AFM stateis {S|t}, which combines the 180◦ rotation of spins S = iσ̂y(σ̂y being the Pauli matrix) with the lattice shift t, connectingantiferromagnetically coupled sublattices. This means that theeigenvectors for the spin states ↑ and ↓ differ only by a phasefactor, which guarantees that (i) the spin bands are degenerateand (ii) the contributions of these bands to the anomalous Hallconductivity are equal to each other and, instead of the partialcancellation, which would occur in regular ferromagnets, wehave an addition of such contributions [28]. The situationis fundamentally different from the IT symmetry realizedin magnetoelectric materials [7,8,12], where the bands arealso degenerate, but the contributions of these bands to AHEcancel each other [29].The time-reversal operation is the combination of S andthe complex conjugation K : T = SK . Therefore, if the mi-croscopic Hamiltonian is complex, the symmetry operation{S|t} is not the same as {T |t}, which is expected for regularantiferromagnets. That is why the spin degeneracy can existeven when the time-reversal symmetry is broken. The spindegeneracy in this case is not the Kramers degeneracy, be-cause the latter implies that the system is T -invariant, whichis obviously not the case here.In this work, we will explicitly show that the {S|t} symme-try is directly related to the fact that I is conserved in thecentrosymmetric antiferromagnets, which imposes a severeconstraint on the form of the SO interaction in magneticbonds. It is true that I transforms each sublattice to itselfand does not interconnect the sites belonging to differentsublattices [30]. However, I will unambiguously specify theproperties of magnetic bonds; namely, knowing the bondproperties around one magnetic site, one can find the prop-erties around a neighboring site, belonging to a differentsublattice. In particular, we will explicitly show that the SO in-teraction on the centrosymmetric antipolar lattice transformsas an AFM object and changes sign upon exchanging thetwo magnetic sublattices. That is why it collaborates withthe AFM Néel field and leads to the {S|t} symmetry of themicroscopic Hamiltonian.There is another important aspect of the {S|t} symmetry:the spin rotation S is combined with the shift t, which canbe viewed as the lattice translation of a more compact unitcell with only one magnetic site. This constitutes the basisof the so-called generalized Bloch theorem [31], which statesthat by a unitary transformation to the local coordinate frame,where all the spins are pointed in the positive direction of z,the AFM system can be described within this compact unitcell as if it were a “ferromagnetic” one. Thus the generalizedBloch theorem provides a mapping of the AFM system ontoan FM one, which naturally explains the breaking of the time-reversal symmetry and the emergence of effects associatedwith it, which are originally known for ferromagnets, but canbe realized in some AFM systems.The rest of the article is organized as follows. In Sec. II,we will consider the basic symmetry properties of SO inter-action imposed by I in combination with lattice translations.Then, in Sec. III, we will show that the behavior of the SOinteractions is related to even more fundamental properties ofthe magnetoelectric coupling in antipolar lattices. Particularly,in order to preserve I, the distortion must be antipolar, whichinevitably leads to the doubling of the unit cell and the sign154413-2HIDDEN FERROMAGNETISM OF CENTROSYMMETRIC … PHYSICAL REVIEW B 114, 154413 (2026)change of the SO interaction when it is considered around twosublattices. Then, in Sec. IV, we will apply the generalizedBloch theorem and show that, for some components of theSO interaction, the sign change can be compensated by thetransformation to the local coordinate frame, so that the AFMsystem can be formally described as an FM one, with a smallerunit cell containing only one magnetic site. In Sec. V, we willintroduce the minimal model for the centrosymmetric antifer-romagnets with broken T and derive transparent expressionsfor the anomalous Hall conductivity and orbital magnetizationin the local coordinate frame. Applications for the squarelattice, monoclinic VF4 and CuF2, as well as RuO2-type mate-rials with tetragonal symmetry will be considered in Sec. VI.In Sec. VII, we will explain how different contributionsto AHE, associated with spin-degenerate and nondegeneratebands, can be evaluated in realistic electronic structure cal-culations. Finally, in Sec. VIII, we will summarize our work.Two Appendixes deal with the unitary transformation of theSO interaction and elimination of the same-sign componentsresponsible for the weak spin ferromagnetism, and construc-tion of the model Hamiltonian for VF4 and CuF2 on the basisof first-principles electronic structure calculations.II. BASIC CONSIDERATIONSLet T1, T2, and T3 be the primitive translations of a crys-tal, consisting of two sublattices, which can be transformedto each other by a rotation, combined with the shift of theorigin by t = 12 (T1 + T2 + T3). Therefore, if one sublatticeis centered at the origin 0 = (0, 0, 0), another sublattice willbe centered at t, and if 0 is an inversion center, t is anotherinversion center [30]. Nevertheless, the midpoint between 0and t is not the inversion center, so that there are finite DMinteractions operating between different sublattices [5,27].The atomic positions in both sublattices can be generatedby the vectors t1 = 12 (−T1 + T2 + T3), t2 = 12 (T1 − T2 +T3), and t3 = 12 (T1 + T2 − T3). However, t1, t2, and t3 donot necessarily transform the DM interactions into themselvesand, in this sense, are not the primitive translations. In anycase, the atomic positions are fully specified by the vectorsR = lt1 + mt2 + nt3. If l + m + n is even, the atom belongsto sublattice 1. If it is odd, the atom belongs to sublattice 2.Thus the DM interactions, DR,R′ , operating between differentsublattices, are finite if R′ − R is an odd superposition of t1,t2, and t3.The DM interactions are induced by the SO coupling. Inthe simplest one-orbital model, the SO interactions are givenby ĤsoR,R′ = itR,R′ · σ̂, where σ̂ is the vector of Pauli matricesand DR,R′ ∼ tR,R′ [32,33]. Therefore tR,R′ and DR,R′ obey thesame symmetry rules.The combination of translational invariance and I imposesa severe constraint on the form of ĤsoR,R′ . I requires that foreach bond R′ − R, there should be the bond R′′ − R in theopposite direction, as explained in Fig. 1, where 1-2′′ is thebond opposite to 1-2 and 1-2′′′ is the bond opposite to 1-2′.Then, I does not change the axial vector tR,R′ and, therefore,tR,R′ = tR,R′′ . Furthermore, each atomic site is surrounded bythe atoms of another sublattice. The atoms in one sublatticeare translated to each other by T1, T2, and T3, which alsoFIG. 1. Parameters of SO interaction around the atoms of twosublattices in the centrosymmetric structure obeying the {C2x|t} sym-metry (a, top) and {C2y|t} symmetry (b, bottom). The atoms ofsublattices 1 and 2 are shown by filled and open circles, respectively.The bond directions, which are defined starting from the central sitein the direction of neighboring sites, are shown by broken arrows.The vectors tR,R′ , attached to these bonds, are shown by the boldgreen arrows. According to this definition, the bonds around sites 1and 2 have opposite directions, which also flip the directions of thevectors tR,R′ attached to these bonds, as shown in the left and rightparts of the figure. The unit cell is shown by the solid red line.translate the vectors tR,R′ around these atoms as shown inFig. 1. Then, since ĤsoR,R′ is the Hermitian matrix, tR,R′ shouldsatisfy the property tR′,R = −tR,R′ . Therefore, if we take R as acentral site and define the directions of the bond starting fromthis R and going to the neighboring site R′, the parameterstR′,R around the site R′ belonging to another sublattice willdiffer from those around R by the sign, as is clearly seenin Fig. 1. Thus, for an arbitrary R′′ = l ′′t1 + m′′t2 + n′′t3, Iimposes the following constraint:tR+R′′,R′+R′′ = (−1)l ′′+m′′+n′′tR,R′ . (1)The remaining uncertainty is how the parameters tR,R′ be-have in the bonds which are not connected by I, such as thebonds 1-2 and 1-2′ in Fig. 1. This depends on other symme-tries. For instance, in the two-dimensional case, the sublatticescan be connected by combining the twofold rotations abouteither x or y with t: {C2x|t} and {C2y|t}, respectively. Thesymmetry operation {C2x|t} ({C2y|t}) will not only changethe signs of the y (x) and z components of tR,R′ , but alsointerchange the sublattices. Therefore, around each site, thex (y) component of tR,R′ will be sign-alternating, while theother two components will have the same sign in all the bonds.The corresponding patterns of tR,R′ are shown in Figs. 1(a)and 1(b). The sign-alternating component is responsible forAHE and orbital magnetism [19,28], while the same-signcomponents give rise to the weak spin ferromagnetism [5,32].Thus it is clear that weak spin ferromagnetism and AHE154413-3I. V. SOLOVYEV PHYSICAL REVIEW B 114, 154413 (2026)have different microscopic origins, and the latter cannot bedescribed in terms of net spin magnetization [18,29,34].III. ANTIPOLAR DISTORTIONSAND WEAK FERROMAGNETISMIn this section, we will elucidate an even more fundamentalreason why the DM interactions for the weak ferromagnetshave a specific pattern shown in Fig. 1, which preserves theinversion symmetry but doubles the unit cell, resulting in twosublattices. We believe that the most fundamental quantity toconsider is magnetoelectric coupling �PR,R′ , relating the crossproduct of spins in the bond to the electric polarization inthe same bond, �PR,R′ = �PR,R′ · [eR × eR′], where eR is the unitvector in the direction of spin at the site R [35–37].�PR,R′ ≡ [Pv,cR,R′] is the 3 × 3 tensor. According to ourconventions [37], the bold character refers to the spin com-ponents (c), which couple to the cross product, while thevector symbol refers to the x, y, and z components (v) ofelectric polarization. Thus, under spatial inversion, the vcomponents will transform according to the normal vectorrules, while the c components will transform according to thepseudovector rule.�PR,R′ can be finite in centrosymmetric bonds. Indeed,the spatial inversion about the center of the bond yieldsthe following property: I �PR,R′ = − �PR′,R = �PR,R′ , whichmeans that �PR,R′ does not necessarily vanish. This is thereason why the electric polarization �P in insulators can beinduced by a noncollinear alignment of spins in otherwisecentrosymmetric crystals [35,36]. Alternatively, the electricfield �ER,R′ , acting in the bond, will interact with �PR,R′ andcauses the spins eR and eR′ to become noncollinear. The cor-responding interaction energy is given by DR,R′ · [eR × eR′],where DR,R′ = − �ER,R′ · �PR,R′ is the DM interaction inducedby the electric field.For insulators, �PR,R′ can be evaluated in terms of thesuperexchange theory, similar to the DM coupling. Indeed,the latter is given by DR,R′ = 2tR,R′ tR,R′U , while the former is�PR,R′ = − 2eVtR,R′ �rR,R′U , where −e is the electron charge, V isthe unit cell volume, U is the on-site Coulomb repulsion,t̂R,R′ = tR,R′ 1̂ + itR,R′ · σ̂ is the transfer integral expanded interms of the 2 × 2 unity matrix 1̂ and the vector of Paulimatrices, and �̂rR,R′ = �rR,R′ 1̂ + i�rR,R′ · σ̂ is a similar expansionfor the position operator [36].�PR,R′ obeys certain symmetry properties. To be specific,let us consider the ideal square lattice, which can be trans-formed to itself by the fourfold rotations around z (C4z) and thespatial inversion about the lattice sites. The primitive transla-tions are t1 and t2, so that each lattice point is specified byR = m t1 + n t2. Furthermore, each bond obeys the twofoldrotational symmetry about itself. Then, for the single bondalong x, say 1-2 in Fig. 2, this tensor will have the followingform [37]:�P1,2 =⎛⎝Px,x 0 00 Py,y Py,z0 P z,x P z,z⎞⎠.FIG. 2. The form of magnetoelectric tensors �PR,R′ in the bondsaround two neighboring sites in the ideal square lattice. The direc-tions of the bonds are indicated by arrows.The bond 1-2 can be transformed to the bond 1-2′′ by I,which changes the sign of the whole magnetoelectric tensor,�P1,2′′ = − �P1,2. Alternatively, 1-2 can be transformed to 1-2′′by the twofold rotation about z, which changes the sign ofonly off-diagonal elements. The combination of these sym-metry properties imposes a constraint on the form of �PR,R′ ,which is summarized in Fig. 2: for each bond, the tensor�PR,R′ is specified by only two elements, Py,z and P z,y. TheKatsura-Nagaosa-Balatsky rule [35], which can be justifiedfor high symmetries of the bond [37], additionally requiresP z,y = −Py,z [35,36]. However, for lower symmetries thisrule does not apply [37], so that generally P z,y and Py,z aretwo independent parameters. Nevertheless, such details arenot important for our consideration.The magnetoelectric tensor is periodic and, therefore,�PR+R′′,R′+R′′ = �PR,R′ for any R′′ (for instance, site 1 in Fig. 2can be moved to site 2, etc.).Next, let us consider possible patterns of electric fieldscaused by internal atomic displacements. For simplicity, weassume that all fields are parallel to z, but can have differentdirections in different bonds. For instance, such fields canbe due to the buckling of the TM-L-TM bonds, where theintermediate ligand (L) sites are displaced parallel to z relativeto the transition-metal (TM) sites. Then, there are three mainpatterns, which are explained in Fig. 3.(a) Polar pattern, where all �ER,R′ are in the positive direc-tion of z. Since �E is the normal vector, the inversionalinvariance would require �ER,R′ = − �ER,R′′ for each R′and R′′ located on the opposite sides of the inver-sion center R. Therefore the polar pattern breaksI. Nevertheless, the displacements are periodic and�ER+R′′,R′+R′′ = �ER,R′ , so as the DM interactions in-duced by them, which have a characteristic whirlingpattern. The DM interactions in this case are respon-sible for the formation of incommensurate spin-spiraltextures or skyrmion lattices [38–40]. Such a situationis realized in magnetic ferroelectrics such as PbVO3[41], BiFeO3 [42], GaV4S8 [43,44], etc.(b) Antipolar pattern, which preserves the translationalinvariance (all fields are transformed to themselvesby the primitive translations t1 and t2), but breaks I.The main symmetry operation in this case is fourfoldrotoinversion. Such a situation is realized (though withthe additional complications) in Ba2CoGe2O7 [45–48]and Ba2CuGe2O7 [49–51]. An interesting aspect of154413-4HIDDEN FERROMAGNETISM OF CENTROSYMMETRIC … PHYSICAL REVIEW B 114, 154413 (2026)FIG. 3. Main patterns of internal electric fields (left) and corre-sponding Dzyaloshinskii-Moriya interactions (right) on the squarelattice: (a) polar, (b) antipolar noncentrosymmetric, and (c) antipolarcentrosymmetric. The doubled unit cell in the latter case is shown bya broken line. xy is the coordinate frame of the regular square lattice.x′y′ is the coordinate frame, which is typically used for the doubledunit cell, such as in Fig. 1.this symmetry is the possibility of realization of an-tiskyrmion textures [52].(c) Antipolar zigzag pattern, satisfying the property�ER+R′′,R′+R′′ = (−1)m′′+n′′ �ER,R′ and doubling the unitcell as explained in Fig. 3(c). Nevertheless, thesedisplacements respect the inversion symmetry, as dothe vectors of DM interactions. As far as the DMinteractions are concerned, the magnetic texture iscommensurate and characterized by a canting ofspins. The typical example is the weak ferromag-netism [5], realized in La2CuO4 and other mate-rials with an orthorhombically distorted perovskitelattice [13,53].To summarize this section, in the weak FM mode, DMinteractions respect the inversion symmetry. This means thatthe lattice is inevitably antipolar, which results in the dou-bling of the unit cell. For two other modes, DM interactionsare translationally invariant, but the inversion symmetry isbroken. Similar behavior is expected for parameters of SOinteractions in the one-orbital model. In insulators, polar andantipolar displacements lead to the realization of, respectively,ferroelectric and antiferroelectric states (so that weak ferro-magnetism is a consequence of antiferroelectricity). In metals,the dipoles induced by atomic displacements are screened byconduction electrons. Nevertheless, the displacements them-selves can persist, and there are many examples of polar andantipolar metallic materials.IV. GENERALIZED BLOCH THEOREMSuppose that two sublattices are ordered antiferromagneti-cally. The corresponding AFM order can be described by thepropagation vector q = Gk , where Gk = 2πV εi jk[Ti × T j] isone of the reciprocal lattice translations (k = 1, 2, or 3) andεi jk is the antisymmetric Levi-Civita symbol, so that q · R = 0and π (mod 2π ) for sublattices 1 and 2, respectively.Then, let us assume that the spins lie in the xy plane,forming the angle α with the axis x. The corresponding Néelfield is given by ±B(cos ασ̂x + sin ασ̂y), where the signs +and − stand for sublattices 1 and 2, respectively.Consider the symmetry operations {ÛR|R}, combining thelattice shift R with the SU(2) rotation of spinsÛR = 1√2(1 −11 1)(ei(q·R+α)2 00 e− i(q·R+α)2), (2)such that, after the transformation, the magnetic field at bothsites is pointed along z, and the corresponding term in theHamiltonian is given by Bσ̂z at both magnetic sublattices.Thus, from the viewpoint of the magnetic field in the localcoordinate frame, the system behaves as a ferromagnet. Theresult is well known in the theory of spin-spiral magnetism[31,54]. Our next goal is to find whether the same appliesto the transfer integrals t̂R,R′ = tR,R′ 1̂ + itR,R′ · σ̂, where thefirst and second terms are, respectively, even and odd in theSO coupling. To the lowest order, tR,R′ is the regular transferintegral without the SO coupling, while tR,R′ is induced by theSO coupling (or simply the SO coupling). Therefore we haveto find how σ̂ and 1̂ are transformed by ÛR and ÛR′ . Thesetransformations are given byÛR σ̂xÛ†R′ = − cos(q · (R + R′)2+ α)σ̂z − sin(q · (R + R′)2+ α)σ̂y, (3)ÛR σ̂yÛ†R′ = − sin(q · (R + R′)2+ α)σ̂z + cos(q · (R + R′)2+ α)σ̂y, (4)ÛR σ̂zÛ†R′ = i sinq · (R − R′)21̂ + cosq · (R − R′)2σ̂x, (5)andÛR1̂Û †R′ = cosq · (R − R′)21̂ + i sinq · (R − R′)2σ̂x. (6)154413-5I. V. SOLOVYEV PHYSICAL REVIEW B 114, 154413 (2026)Considering the transfer integrals and the SO interactionbetween different sublattices, we have q·(R±R′ )2 = π2 (mod π )and, therefore,ÛR σ̂xÛ†R′ = − sinq · (R + R′)2(cos ασ̂y − sin ασ̂z ), (7)ÛR σ̂yÛ†R′ = − sinq · (R + R′)2(sin ασ̂y + cos ασ̂z ), (8)ÛR σ̂zÛ†R′ = i sinq · (R − R′)21̂, (9)andÛR1̂Û †R′ = i sinq · (R − R′)2σ̂x. (10)Thus we have the following property: ÛR+R′′ 1̂Û †R′+R′′ =ÛR1̂Û †R′ for any R′′. The same holds for ÛR σ̂zÛ†R′ . On theother hand, for a = x and y, we have ÛR+R′′ σ̂aÛ†R′+R′′ =(−1)l ′′+m′′+n′′ÛR σ̂aÛ†R′ , which is the same as the property ofthe SO interaction parameters tR,R′ given by Eq. (1).Combining these transformations of 1̂ and σ̂ with tR,R′ andtR,R′ = (t xR,R′ , t yR,R′ , t zR,R′ ), it is straightforward to see thattR+R′′,R′+R′′ ÛR+R′′ 1̂Û †R′+R′′ = tR,R′ ÛR1̂Û †R′ , (11)t xR+R′′,R′+R′′ ÛR+R′′ σ̂xÛ†R′+R′′ = t xR,R′ ÛR σ̂xÛ†R′ , (12)t yR+R′′,R′+R′′ ÛR+R′′ σ̂yÛ†R′+R′′ = t yR,R′ ÛR σ̂yÛ†R′ , (13)andt zR+R′′,R′+R′′ ÛR+R′′ σ̂zÛ†R′+R′′ = (−1)l ′′+m′′+n′′t zR,R′ ÛR σ̂zÛ†R′ .(14)Thus, in the local coordinate frame, not only regular trans-fer integrals, but also two components of the SO interactionappear to be periodic on the lattice specified by the trans-lations t1, t2, and t3, and transform sublattices 1 and 2 intoeach other. The third (z) component of the SO interactionsis not periodic on this compact lattice and would generallyrequire the use of the two-sublattice unit cell specified byT1, T2, and T3. However, for the weak ferromagnets, atleast one component of the SO interaction should have thesame sign in all the bonds around each magnetic site and,therefore, can be eliminated via a unitary transformation onone of the sublattices [28,32,55]. By properly specifying thequantization axis, this component can always be chosen as z.The elimination procedure, which is valid to first order in theSO coupling, is explained in Appendix A. In some systems,like RuO2 and La2CuO4, having, respectively, P42/mnm andBmab symmetry, the z component of the SO coupling is equalto zero and, therefore, no elimination is required. At the sametime, the y component of the SO coupling in La2CuO4, whichis responsible for the weak ferromagnetism along z, shouldbe eliminated. On the other hand, the sign-alternating com-ponents of the SO coupling do not contribute to the weakferromagnetism and cannot be eliminated simultaneously inall the bonds [28].For the transfer integrals operating within the sublattices,the transformation to the local coordinate frame is given byÛR1̂Û †R′ = cosq · (R − R′)21̂, (15)which is the same for sublattices 1 and 2. Therefore thesesublattices can be transformed to each other by the translationst1, t2, and t3 if the transfer integrals tR,R′ operating in sublattice1 are the same as in sublattice 2. In this case, there will be noaltermagnetic band splitting, originating from the differenceof tR,R′ in sublattices 1 and 2 [28,56]. However, this splittingdoes not seem to play a key role in AHE, which emergesalready in the spin-degenerate bands [19,28]. Therefore, as thefirst approximation, the change of the transfer integrals relatedto the altermagnetic band splitting can be neglected (an alter-native scheme, which can be used in first-principles electronicstructure calculations, will be considered in Sec. VII).To summarize this section, the minimal model for AHE incentrosymmetric antiferromagnets can be formulated in thelocal coordinate frame, combining the translations t1, t2, andt3 with the SU(2) rotations of spins. In centrosymmetric anti-ferromagnets, the transformation to the local coordinate framecompensates the effect of antipolar displacements, which re-sulted in the doubling of the unit cell (see Sec. III). Theelectronic Hamiltonian in this local coordinate frame becomesperiodic on the lattice specified by t1, t2, and t3. Thus the AFMsystem with the SO coupling can be effectively described asan FM one with only one magnetic site in the cell.V. MINIMAL MODELAHE in centrosymmetric antiferromagnets was predictedseveral decades ago on phenomenological grounds [8,9]. In1997, the magneto-optical effect (the ac analog of AHE) inweak ferromagnets was studied quantitatively on the basis offirst-principles electronic structure calculations [34]. In par-ticular, it was argued that the effect is strong and comparablewith the one in the regular FM state (though the net spinmagnetization in the weak and regular FM states differed bytwo orders of magnitude). This behavior was attributed to theform of orbital magnetization, which substantially deviatesfrom the spin one. The imaginary antisymmetric part of op-tical conductivity, Im[σ A(ω)], which was actually consideredin Ref. [34], is related to the real part, Re[σ A(ω)], by theKramers-Kronig transformation, meaning that if Im[σ A(ω)]is finite, Re[σ A(ω)] is also finite. Thus the conclusions ofRef. [34] have direct implications for the behavior of AHEin the metallic regime. In this section, we introduce the mini-mal model, describing AHE and orbital magnetization on themicroscopic level.A. Hamiltonian1. Global coordinate frameMain parameters of the model Hamiltonian are summa-rized in Fig. 4 for the square lattice [28,56].They includehoppings between first (t1), second (t2), and third (t3) nearestneighbors. δt2 describes the orthorhombic distortion, whichmakes the directions x and y inequivalent. δt3 is responsiblefor the altermagnetic splitting of bands. δt2 and δt3 operate154413-6HIDDEN FERROMAGNETISM OF CENTROSYMMETRIC … PHYSICAL REVIEW B 114, 154413 (2026)FIG. 4. Main parameters of the model Hamiltonian for the squarelattice: (a) hoppings between first nearest neighbors; (b) hoppingsbetween second and third nearest neighbors; and (c) vectors of spin-orbit interaction (shown by bold arrows) around magnetic sites 1 and2 after eliminating weakly ferromagnetic components. The directionsof the bonds are shown by dashed arrows.within the sublattices. Nevertheless, δt2 has the same formin sublattices 1 and 2, while δt3 alternates as explained inFig. 4(b). It is assumed that the SO interaction has only onesign-alternating component x. Two other components have thesame sign in all the bonds and can be eliminated as explainedin Appendix A. The Néel field is also parallel to x. Then, theSO interaction and Néel field can be realigned along z, whichcorresponds to the global rotation of spins. The corresponding4 × 4 spin-diagonal Hamiltonian is given by [28]Ĥgk = hk1̂ − δh3kτ̂z + h1kτ̂x − Bτ̂zσ̂z + hsok τ̂yσ̂z, (16)where τ̂ = (τ̂x, τ̂y, τ̂z ) is a pseudospin describing the inter-action between two AFM sublattices [56], and hk, δh3k, andhsok are the Fourier images of corresponding parameters inreal space [28,57]. The corresponding eigenvalues are givenby εσk,ν = hk + ν√(σB + δh3k)2 + (h1k)2 + (hsok )2, where ν =± and σ = ± are band and spin indices, respectively.Hamiltonian (16) constitutes the minimal model for al-termagnets, which, with some variations, is widely used asthe toy model for materials with different crystallographicsymmetries [17–19,28,56]. Nevertheless, below we will showthat, as far as the AHE is concerned, there is an even morecompact model describing this phenomenon in centrosymmet-ric antiferromagnets.It is instructive to consider how Ĥgk breaks time-reversalsymmetry. The transformation rules for the different termsof Hamiltonian (16) under the symmetry operations K , T ,S , {T |t}, and {S|t}, where t = ( 12 , 12 ) is the shift connectingtwo magnetic sublattices, are summarized in Table I. There isno symmetry, which would transform all four terms to them-selves. Thus all symmetries are broken: not only T , but alsoits components K and S , both alone and in the combinationwith t. For instance, the Néel field breaks T and S , the SOinteraction breaks K , S and {T |t}, while the altermagneticsplitting breaks {T |t} and {S|t}. Nevertheless, there are twointeresting possibilities, where only one of the symmetries,either K or {S|t}, is broken, while another one is not. Thismeans that the {T |t} symmetry is automatically broken, butTABLE I. Transformation of different terms of Hamiltonian (16)by the symmetry operations K , S, T , {S|t}, and {T |t}. The terms τ̂x ,τ̂z, τ̂yσ̂z, and τ̂zσ̂z stand for, respectively, nearest-neighbor hoppings,altermagnetic band splitting, spin-orbit interaction, and Néel field.Operation τ̂x τ̂z τ̂yσ̂z τ̂zσ̂zK τ̂x τ̂z −τ̂yσ̂z τ̂zσ̂zS τ̂x τ̂z −τ̂yσ̂z −τ̂zσ̂zT τ̂x τ̂z τ̂yσ̂z −τ̂zσ̂z{S|t} τ̂x −τ̂z τ̂yσ̂z τ̂zσ̂z{T |t} τ̂x −τ̂z −τ̂yσ̂z τ̂zσ̂zthe system still obeys certain symmetry properties governedby either K or {S|t}.(1) The first scenario is widely discussed in the con-text of altermagnetism, which typically starts with theanalysis of spin splitting of bands without relativisticSO interaction [2,3,25]. Then, the altermagnetic termbreaks {S|t}, so that the {T |t} symmetry appears to bebroken even without SO interaction. Nevertheless, theHamiltonian is real (K invariant). Therefore AHE and netorbital magnetization will vanish. Besides a regular AFMorder parameter, such a state can be described in terms offerroically ordered spin magnetic octupoles [10,58–60].(2) The second scenario is realized when the SO couplingis finite, but the altermagnetic splitting term identicallyvanishes. Then, the SO interaction, which is complex,breaks K . Therefore the {T |t} symmetry will be alsobroken. Nevertheless, the Hamiltonian remains invariantunder {S|t}. Since K is broken, the system will exhibitAHE and net orbital magnetization, manifesting ferro-type orbital magnetic dipolar order.In this work, we explore the second possibility, whichis related to AHE. In a narrow sense, the condition δh3k =0 in Eq. (16) defines altermagnetic nodal surfaces of spin-degenerate bands without SO coupling [18,56,61]. The AHEis induced by the sign-alternating component of the SO cou-pling, which should be parallel to the Néel field, as in Eq. (16)[19,28]. However, in this configuration, the SO interactiondoes not lift the spin degeneracy [56]. Thus altermagneticnodal surfaces are not essential for the AHE in centrosym-metric antiferromagnets. Moreover, there are many situationswhen δt3 itself is small or equal to zero, either accidentallyor by symmetry. This leads to the massive spin degeneracy ofbands throughout the Brillouin zone even in the presence ofrelativistic SO coupling [19,28].2. Local coordinate frameUsing the generalized Bloch theorem, the minimal modelfor AHE in the reciprocal space can be formulated in the localcoordinate frame via the Fourier transform, which combinesthe Bloch factor, eik·R, associated with the translation R, withadditional prefactors arising from the SU(2) rotation of spins,ÛR , with the propagation vector q = Gk [31,54]. Supposethat one component of the SO coupling is sign-alternating,while two other components have the same sign and can be154413-7I. V. SOLOVYEV PHYSICAL REVIEW B 114, 154413 (2026)eliminated, as explained in Appendix A. As it will becomeclear below, this is a rather general situation, relevant to thebehavior of many weak ferromagnets. The situation wherethere are two sign-alternating components and only one com-ponent with the same sign (or equal to zero) is also possible.It will be considered in Sec. VI C.The 2 × 2 Hamiltonian describing a two-component weakferromagnet is given byĤk = hk1̂ + h1kσ̂x − hykσ̂y + Bσ̂z, (17)where h1k and hyk are the Fourier images of the nearest-neighborhoppings and sign-alternating part of the SO coupling, respec-tively, and hk is the Fourier image of transfer integrals withinthe sublattice, assuming that these transfer integrals have thesame form in both AFM sublattices and the altermagneticpart, responsible for the spin splitting, can be neglected. TheseFourier images can be obtained using Eqs. (10), (8), and (15),respectively. The Néel field is taken in the direction of thesign-alternating part of the SO coupling. The directions x andy correspond, respectively, to α = 0 and π/2 in Eqs. (7) and(8). In both cases, the SO interaction is described by the matrixσ̂y in the local coordinate frame, as reflected in Eq. (17).According to Eq. (10), the nearest-neighbor hoppings are de-scribed by the matrix σ̂x. Finally, the magnetic field in thelocal coordinate frame is aligned along z at both magneticsites.After replacing the Pauli matrices σ by pseudospin matri-ces τ, Ĥk is totally equivalent to the Hamiltonian (16) in theglobal coordinate frame for the σ = − states and δt3 = 0. Aswill be argued in Sec. VI A, the parameters hk, h1k, and hykof these two Hamiltonians also coincide. Obviously, it willyield the same anomalous Hall conductivity and net orbitalmagnetization.The Hamiltonian has two eigenvalues:ε±k = hk ∓√B2 + (h1k)2 + (hyk)2 ≡ hk ∓ Ak, (18)for the majority (ε+k ) and minority (ε−k ) spin states in the localcoordinate frame. Seeking the corresponding eigenvectors inthe form|u+k 〉 =(cos θkeiφksin θk)≡ |uk 〉 (19)and|u−k 〉 =(− sin θkcos θke−iφk), (20)it is straightforward to find that−Ak cos 2θk = B (21)and−Ak sin 2θkeiφk = h1k + ihyk, (22)whereθk = 12arctan√(h1k)2 + (hyk)2B(23)(0 � θk < π ) andφk = arctan(hykh1k)(24)(0 � φk < 2π ).B. Anomalous Hall conductivityUsing the above expressions and considering the situationwhen B is sufficiently large, so that the majority-spin band ispartly occupied, while the minority-spin band is empty, it isstraightforward to obtain that the Berry curvature, 
c(k) =−2Im〈∂ka uk|∂kbuk〉 [62], becomes
ck = sin 2θk(∂kaθk∂kbφk − ∂kaφk∂kbθk), (25)wheresin 2θk = −√(h1k)2 + (hyk)2Ak, (26)∂kaθk = 12(h1k∂ka h1k + hyk∂ka hyk)BA2k√(h1k)2 + (hyk)2, (27)∂kaφk = h1k∂ka hyk − hyk∂ka h1k(h1k)2 + (hyk)2 , (28)and abc is an even permutation of xyz.Then, after some algebra, 
ck can be further rearranged as
ck = B2A3k(∂ka hyk∂kbh1k − ∂ka h1k∂kbhyk). (29)The anomalous Hall conductivity is given by the Brillouinzone (BZ) integral [62]σab = − 1V∫BZdk
BZfkck,where 
BZ = (2π )3V is the BZ volume and fk is the Fermi-Dirac distribution function for ε+k . In the large B limit, 
ck ∼1B|B| [28]. Therefore the Taylor expansion of σab in terms of Bis not meaningful and only the expansion in the directions ofthe Néel field is generally well defined [63]. However, it doesnot contradict phenomenological theories [5–10,12], wherethe absolute value of the AFM order parameter (and otherquantities) is generally unknown, and these theories establishonly the directional dependence.The magneto-optical effect (the ac analog of AHE) can bealso evaluated in the local coordinate frame by considering theoptical transitions between the u−k and u+k states.C. Orbital magnetizationThe orbital magnetization along c is given by the BZintegral ofMck = Im〈∂ka uk|Ĥk + ε+k − 2εF|∂kbuk〉 (30)154413-8HIDDEN FERROMAGNETISM OF CENTROSYMMETRIC … PHYSICAL REVIEW B 114, 154413 (2026)with the Fermi-Dirac distribution function [64,65]. Using theexplicit form of Ĥk, ε+k , and |uk〉, Mck can be rearranged asthe sum of four contributions [28]:Mck,I = −12sin 2θk(2hk − Ak)(∂kaθk∂kbφk − ∂kaφk∂kbθk),(31)Mck,II =(εF − B2)
ck, (32)Mck,III = h1k cos2 θk cos φk(∂kaθk∂kbφk − ∂kaφk∂kbθk), (33)andMck,IV = hyk cos2 θk sin φk(∂kaθk∂kbφk − ∂kaφk∂kbθk). (34)Then, using Eq. (25), all these terms can be related to 
ckasMck,I = −12(2hk − Ak)
ck, (35)Mck,III = h1k cos φk cos2 θksin 2θk
ck, (36)andMck,IV = hyk sin φk cos2 θksin 2θk
ck. (37)Noting thatcos2 θksin 2θk= 12B − Ak√(h1k)2 + (hyk)2(38)and h1k cos φk + hyk sin φk =√(h1k)2 + (hyk)2, it is straightfor-ward to find that Mck,III + Mck,IV = 12 (B − Ak)
ck. Summingup all four terms, one can get the final expression:Mck = (εF − hk )
ck. (39)Thus the orbital magnetization isMc =∫BZdk
BZfk(εF − hk )
ck.VI. EXAMPLESA. Square perovskite latticeAs the first example, let us consider the single layer of or-thorhombically distorted perovskites with the Pbnm or Bmabsymmetry [28]. The lattice is specified by the primitive trans-lations T1 = (1, 0, 0) and T2 = (0, 1, 0), and there are twosublattices, which are connected by the vectors t1 = ( 12 , 12 ) ≡t and t2 = (− 12 , 12 ). The symmetry operation transformingthe two sublattices into each other is {C2x|t}, meaning that xcomponents of the SO coupling are sign-alternating. For thespace group Bmab, the lattice is additionally invariant underthe mirror reflection y → −y.The minimal model was introduced in Sec. V A 1 andexplained in Fig. 4. We would like to emphasize again thatthe SO interaction obeys the following property, being theconsequence of the inversional invariance: ĤsoR+R′′,R′+R′′ =(−1)m′′+n′′ ĤsoR,R′ . In the global frame, the SO interaction isnot periodic on the lattice, specified by t1 and t2. However,it can be made periodic in the local frame as explained inFIG. 5. Results for the square lattice model with the parameters(unless it is specified otherwise) t1 = −1, t2 = −t3 = 0.1, and δt2 =−tx = 0.05 (all are in units of B): (a) Berry curvature, 
zk, in the firstBrillouin zone; (b) similar plot for orbital magnetization, Mzk, cor-responding to nel = 0.5 electrons; band filling dependence of (c) theanomalous Hall conductivity, σxy, and (d) the orbital magnetization,Mz; (e) Fermi surface at nel = 0.5 with (red) and without (blue) theorthorhombic strain δt2; and (f) the orthorhombic strain dependenceof σxy and Mz for nel = 0.9.Secs. II–IV. We do not consider the altermagnetic deforma-tion of the third-neighbor hoppings, which is identically equalto zero for the Bmab symmetry of La2CuO4 and expected tobe small for other materials [28]. For La2CuO4, this is the con-sequence of the mirror reflection y → −y, which makes thebonds with t3 ± δt3 in Fig. 4(b) equivalent. For the one-orbitalmodel, which is widely used in the theory of superconductiv-ity [66,67], it leads to δt3 = 0. In a more general multiorbitalmodel, δt3 can be finite, provided that it operates betweendifferent orbitals belonging to odd and even representationsof the point group. Nevertheless, the band splitting, which isdriven by this δt3, is small [2].Thus one can readily find the Hamiltonian (17) in thelocal coordinate frame, where hk, h1k, and hyk are obtained bycombining the Fourier transforms with the matrix elements of1̂ and σ̂x given by Eqs. (10) and (7), respectively. Moreover, itis convenient to shift the k mesh: (kx, ky) → (kx + π, ky). Al-together, this yields hk = 2t2(cos kx + cos ky) + 2δt2(cos kx −cos ky) + 4t3 cos kx cos ky, h1k = 4t1 cos kx2 cos ky2 , and hyk =−4tx sin kx2 sin ky2 , which coincide with the parameters reportedin Ref. [28] in the global coordinate frame [57].Berry curvature can be written in the compact form, usingEq. (29):
zk = Bt1txA3k(cos kx − cos ky). (40)The results are summarized in Fig. 5. In comparison withcalculations in the global coordinate frame [28], the gener-alized Bloch theorem allows us to use a smaller unit cell.Therefore the first Brillouin zone becomes larger, where 
zkhas two positive and two negative areas [Fig. 5(a)], which tendto compensate each other. Therefore, in order to obtain finiteσxy and Mz, it is essential to consider the orthorhombic strain,δt2 [19,28]. Although the Berry curvature does not depend onδt2, the latter deforms the Fermi surface, as shown in Fig. 5(e),154413-9I. V. SOLOVYEV PHYSICAL REVIEW B 114, 154413 (2026)so that the positive and negative areas of 
zk will contribute toσxy with different weights, resulting in finite σxy. According toEq. (39), in the case of Mz, there is an additional modulationof 
zk by hk, which makes inequivalent the positive and neg-ative areas of Mzk [see Fig. 5(b)]. Nevertheless, the effect issmall if δt2 is small.Finally, AHE and orbital magnetization can be efficientlycontrolled by the orthorhombic strain, where the transitionfrom tensile (δt2 < 0) to compressive (δt2 > 0) strain changesthe sign of σxy and Mz [see Fig. 5(f)]. This effect has a clearsimilarity with piezomagnetism [20,24,28]. For small δt2, thecontribution εFzk to the orbital magnetization dominates andMz ∼ σxy.B. VF4 and CuF2As the next example, we consider the minimal model forVF4 and CuF2, which are regarded as altermagnetic can-didates [25]. Both materials crystallize in the monoclinicstructure (the space group P21/c, No. 14) with two for-mula units in the primitive cell [68,69]. V4+ has one delectron, while Cu2+ has one d hole. According to the local-density approximation (LDA), the electronic structure of VF4and CuF2 near the Fermi level is characterized by a well-isolated half-filled band. Therefore, in the first approximation,the properties of these materials can be described by theone-orbital model with the parameters derived from the first-principles calculations. This is basically an extension of theprevious considerations to the case of lower crystallographicsymmetry. The details of electronic structure calculations andconstruction of the model are described in Appendix B. Themain results are summarized in Table II.The primitive translations are T1 = (a sin β, 0, a cos β ),T2 = (0, b, 0), and T3 = (0, 0, c), both for VF4 and CuF2.Two sublattices are transformed into each other by the sym-metry operation {C2y|t}. Therefore the x and z componentsof the SO interaction between the nearest neighbors havethe same sign in all the bonds and can be eliminated (seeAppendix A). The y component is sign-alternating. The Néelfield parallel to the y axis will result in weak ferromagnetismalong x and z.The atoms in the nearest neighborhood of each atomicsite are located at t3, −t3, t3 − T2, and −t3 + T2. The AFMpropagation vector can be taken as q = G2, so that sin q·t32 = 1and sin q·(t3−T2 )2 = −1. Furthermore, in the first two bonds, theSO coupling is ity, while in the other two bonds it is −ity.Then, using Eqs. (10) and (8) for α = π/2, and shifting the k-mesh k → k + q2 , one can find h1k = 4t1 cos k·(T1−T3 )2 cos k·T22TABLE II. Experimental lattice parameters (a, b, and c are inÅ, β is in degrees) and main parameters of the one-orbital modelfor VF4 and CuF2: the nearest-neighbor hopping t1 (in meV), thesign-alternating component of the spin-orbit coupling ty (in meV),and on-site Coulomb repulsion U (in eV). The lattice parameters aretaken from Refs. [68, 69].a b c β t1 ty UVF4 5.381 5.170 5.340 59.74 −100.66 0.71 3.2CuF2 3.297 4.562 4.616 83.29 −173.73 5.02 4.0and hyk = 4ty sin k·(T1−T3 )2 sin k·T22 . Finally, hk in Eq. (17) isthe Fourier image of transfer integrals between atoms of onemagnetic sublattice, and assuming that they are the same inboth sublattices (otherwise the generalized Bloch theoremdoes not apply). Technically, hk is obtained by averagingthe transfer integrals over two magnetic sublattices. The de-tails and applicability of this approximation are discussed inAppendix B. Briefly, the hopping parameters δtR,R′ responsi-ble for the altermagnetic spin splitting of bands are small inVF4, but can be comparable with regular hoppings tR,R′ in thecase of CuF2. In the strong coupling limit, the contributionof degenerate bands to AHE is of the order of t1tyB|B| , whereB ≈ U2 . The spin splitting yields the additional contribution∼ 2t1tyδtR,R′|B|3 , which is smaller by the factor2δtR,R′|B| [28]. Therealistic parameters |δtR,R′ | < 30 meV (Appendix B) and U =4 eV (Table II) yield the following estimate2|δtR,R′ ||B| < 0.03.Therefore the altermagnetic contribution to AHE, originatingfrom the spin splitting of bands, is expected to be small.Then, using Eq. (29), it is straightforward to find that
xk = Bt1tyA3k(a cos β − c)b{cos k · (T1 − T3) − cos k · T2},(41)yk = 0, and
zk = −Bt1tyA3kab sin β{cos k · (T1 − T3) − cos k · T2}. (42)Therefore the ratio
xk
zk= c − a cos βa sin β(43)is controlled by the geometrical factor, depending only on thelattice parameters.These tendencies are clearly seen in the behavior of AHEand orbital magnetization (Fig. 6). The nonvanishing yz andxy components of the Hall conductivity are related to eachother by a scaling transformation, which is expected fromEq. (43). The same holds for the x and z components ofthe orbital magnetization. Furthermore, the shape of orbitalmagnetization basically repeats the one of AHE, as expectedfrom Eq. (39).Finally, we would like to comment on the possibility ofpractical realization of AHE (under doping) or magneto-optical effect in VF4 and CuF2. One important question iswhether q = G2 ≡ (0, 1, 0) corresponds to the true magneticground states of VF4 and CuF2. For instance, from the view-point of symmetry, a similar material CuO could be classifiedas a potential altermagnet: the space group is C2/c, wherethere are two Cu sublattices. Each sublattice is transformed toitself by I, while different sublattices are transformed to eachother by {C2y|t}. However, the magnetic ground state of CuOcorresponds to q ≈ ( 12 , 0, 12 ), meaning that it is a noncollinearmultiferroic [70]. This tendency is successfully explained bythe behavior of superexchange interactions [71]. The samesituation is expected in VF4 and CuF2, where the analysisof superexchange interactions also predicts the multiferroicq ≈ ( 12 , 0, 12 ) ground state.154413-10HIDDEN FERROMAGNETISM OF CENTROSYMMETRIC … PHYSICAL REVIEW B 114, 154413 (2026)FIG. 6. Band filling dependence of anomalous Hall conductivityand orbital magnetization in VF4 and CuF2.C. RuO2-type systemsRuO2 is viewed as a canonical altermagnetic material [18].It crystallizes in the rutile-type body-centered tetragonal (bct)structure with two sublattices (the space group P42/mnm, No.136). The atomic positions in both sublattices can be formallygenerated by the vectors t1 = (− 12 , 12 , 12 ), t2 = ( 12 ,− 12 , 12 ),and t3 = ( 12 , 12 ,− 12 ), in units of a and c (c/a < 1 being thetetragonal distortion along z). The additional complicationarising in this case is that there are two sign-alternating com-ponents of the SO coupling in the nearest bonds: x and y,as explained in Fig. 7. The third (z) component is identicallyequal to zero. Thus, formally speaking, there is no weak spinferromagnetism arising from DM interactions, which is be-lieved to be a unique aspect of RuO2 and other AFM materialswith the P42/mnm symmetry. However, in other materials,the weakly FM component can be eliminated in the localFIG. 7. Spin-orbit interactions tR,R′ (denoted by bold blue ar-rows) around two magnetic sites in the P42/mnm structure. Thevectors tR,R′ replicate the form of Dzyaloshinskii-Moriya interac-tions. The corresponding bond directions are shown by gray arrows.coordinate frame (see Appendix A). Then, from this point ofview, there is no fundamental difference between RuO2 andother weak ferromagnets.Nevertheless, the weak spin ferromagnetism in rutilecompounds is allowed by symmetry [72] and the micro-scopic mechanism behind it is believed to be the single-ionanisotropy [73]. However, the single-ion anisotropy appearsonly at second order in the SO coupling. Therefore, we expectit to play only a minor role in the AHE and in the orbitalmagnetization, both of which arise already at first order.We assume that the Néel field is parallel to x. Then, accord-ing to Eqs. (7) and (8), in the local coordinate frame specifiedby q = G1 and α = 0, the SO interaction is described by thePauli matrices σ̂y and σ̂z. Other transformations are given byEqs. (10) and (15).Therefore we will deal with the following Hamiltonian:Ĥk = hk1̂ + h1kσ̂x − hykσ̂y + (B + hzk)σ̂z, (44)where it is again convenient to shift the k mesh: k →k + (π, 0, 0). Then, the parameters of Eq. (44) will be givenby hk = 2(t2 − δt2)(cos kx + cos ky) + 2(t2 + δt2) cos kz,h1k = 8t1 cos kx2 cos ky2 cos kz2 , hyk = −8tx sin kx2 cos ky2 sin kz2 , andhzk = 8ty cos kx2 sin ky2 cos kz2 , where t1 is the nearest-neighborhopping between the sublattices and t2 ± δt2 are the hoppingswithin the sublattices: along z (t2 + δt2) and in the xyplane (t2 − δt2). Although for the P42/mnm symmetrytx = ty, it is convenient to treat tx and ty as independent SOparameters. The eigenvalues, eigenfunctions, and propertiesof Hamiltonian (44) can be obtained along the same line as inSec. V after replacing B by B + hzk.Furthermore, we have the properties h−k = hk, h1−k = h1k,hy−k = hyk, and hz−k = −hzk, which immediately yield ε±k (B) =ε±−k(−B) and ε±k (B, ty) = ε±k (−B,−ty). Then, the Berry cur-vature satisfies the following properties: yk(B) = −y−k(−B)(∂kaφk changes the sign for a = z and x, while sin 2θk and∂kaθk do not change the sign) and yk(B, ty) = −yk(−B,−ty )(∂kaθk changes the sign, while sin 2θk and ∂kaφk do not changethe sign). Therefore the Hall conductivity, σzx, will be oddin B, σzx(−B) = −σzx (B), and even in ty, σzx(−ty) = σzx(ty).Moreover, σzx is odd in tx, i.e., when the SO coupling is in thedirection of the Néel field: σzx(−tx ) = −σzx(tx ). Thus, to cal-culate σzx to first order in the SO coupling, ty can be neglected.This is consistent with the general strategy for calculating DMinteractions using the response theory: in order to calculatesome particular (say, x) component of the DM vectors, it issufficient to align the exchange field along x and consider onlythe x component of the SO coupling [33,74].After neglecting ty, the Berry curvature can be obtainedfrom Eq. (29) asyk = 4t1txBA3kcos2 ky2(cos kz − cos kx ). (45)The results are summarized in Fig. 8, using parameters re-ported in Ref. [56]. The largest contributions to yk arisefrom the (2π, 0, 0) → (0, 0, 0) → (0, 0, 2π ) directions in theky = 0 plane, which are additionally modulated along ky,as described by Eq. (45). Moreover, yk has nodal linesalong kz = ±kx. Therefore, if the bands along (0, 0, 0) →(2π, 0, 0) and (0, 0, 0) → (0, 0, 2π ) were equally populated,154413-11I. V. SOLOVYEV PHYSICAL REVIEW B 114, 154413 (2026)FIG. 8. Results for RuO2-type systems with the parameters t1 =1, t2 = 1.53, δt2 = 1.76, and tx = 0.1 reported in Ref. [56], andB/t1 = 8: (a) Berry curvature, (b) orbital magnetization for nel = 0.5electrons, and (c) band dispersion along high-symmetry directionsof the bct Brillouin zone (the Fermi level for nel = 0.5 is shown bya dash-dotted line). Corresponding contour plots of (d) Berry curva-ture and (e) orbital magnetization in the ky = 0 plane (0 � kx � 2πand 0 � kz � 2π ); (f) Band filling dependence of anomalous Hallconductivity and orbital magnetization.σxy would vanish. That is why we need the “tetragonalstrain” δt2 [19,28], which is associated with the tetragonaldistortion c/a < 1 of the bct lattice and leads to consider-ably stronger dispersion along (0, 0, 0) → (0, 0, 2π ). Then,for the nel = 0.5 electrons, the “majority-spin” band is fullypopulated along (0, 0, 0) → (2π, 0, 0) and only partially pop-ulated along (0, 0, 0) → (0, 0, 2π ), resulting in finite σxy. Inthe expression for the orbital magnetization, Eq. (39), ykis additionally modulated by hk, thus destroying the perfectcancellation between the directions (0, 0, 0) → (2π, 0, 0) and(0, 0, 0) → (0, 0, 2π ). Unlike in the previous examples, theform of orbital magnetization substantially deviates from thatof anomalous Hall conductivity, as is clearly seen in Fig. 8(f).This effect is related to the much stronger tetragonal distor-tion, δt2/t1 = 1.76 [56], when the term hkyk in the orbitalmagnetization starts to play a sizable role.VII. CLASSIFICATION BASED ON REALISTICELECTRONIC STRUCTUREThe existence of AHE without band splitting is a relativelynew and important finding expected from the analysis of asimple one-orbital toy model [19,28]. This model can be jus-tified for a limited number of compounds, like κ-type organicconductors [19], La2CuO4 [28], VF4, and CuF2, where theelectronic structure near the Fermi level is indeed charac-terized by a single well-separated band, which can be takenas the basis for the construction of a realistic model withthe input parameters derived from first-principles electronic-structure calculations [66,67,75,76]. This model is typicallyin the strong coupling limit, meaning that the band splitting,ε↑k − ε↓k = δεk, is much weaker than B, so that the contribu-tion to the Berry curvature associated with the band splittingis about δεk/B times smaller than the one coming from degen-erate bands [28].The next important question is how the contributionsrelated to the band splitting and spin degeneracy can beseparated in realistic electronic structure calculations. Forthese purposes, one can use results of the model analy-sis as guidance. First, it is important to stay in the globalcoordinate frame: although the generalized Bloch theoremis vital for clarifying fundamental aspects of time-reversalsymmetry breaking in centrosymmetric antiferromagnets, itrelies on additional approximations, such as the neglect of thespin splitting of AFM bands, which do not necessarily holdin realistic calculations. Furthermore, the generalized Blochtheorem does not provide a substantial technical advantagefor calculating AFM spin structures (in comparison with in-commensurate spin spirals). Second, spin should be a goodquantum number (otherwise, the analysis of spin splitting ismeaningless). Therefore one should keep only one componentof the SO coupling, which is parallel to the Néel field. Thisis similar to what we did in the model. Two other compo-nents of the SO coupling cannot be fully eliminated as in theone-orbital case. However, if the spin canting is small, theircontribution to AHE is also expected to be small [18].Suppose that the crystal is oriented in such a way that theNéel field and SO coupling are parallel to z. Importantly, usingthis setting and only the z component of the SO coupling, weshould be able to reproduce the z component of DM interac-tions [33,74] and, hopefully, the Hall conductivity σxy, whichis related to the sign-alternating part of these interactions [28].The approximation holds to first order in the SO coupling[33,74]. Since the Berry curvature is proportional to the SOcoupling at lowest order [19,28,56], it should be captured bythe present approach.Finally, let us assume that the altermagnetic bands nearthe Fermi level can be classified by the spin-doublet quantumnumber n and spin indices, ↑ or ↓, describing the splitting ofeach such doublet. Then, the Hall conductivity,σxy = − 1V∑n∫BZdk
BZ(f ↑nk↑,znk + f ↓nk↓,znk),can be identically rearranged asσxy = − 1V∑n∫BZdk
BZ(f enko,znk + f onke,znk),where 
o,znk = 12 (
↑,znk − ↓,znk ) and f onk = ( f ↑nk − f ↓nk) are oddin the Néel field B (the interchange of ↑ and ↓), while 
e,znk =12 (
↑,znk + ↓,znk ) and f enk = ( f ↑nk + f ↓nk) are even. Obviously,total σxy is odd in B. Basically, this is a generalization of theone-orbital model [28]. The first term can be finite even whenthe bands are spin-degenerate, while the second term is finiteonly when the ↑ and ↓ bands are split across the Fermi level,154413-12HIDDEN FERROMAGNETISM OF CENTROSYMMETRIC … PHYSICAL REVIEW B 114, 154413 (2026)like in the FM state. Indeed, according to Eq. (29), 
znk is oddin B when the bands are spin-degenerate (in the global frame),manifesting the {S|t} symmetry. The even term, 
e,znk , appearswhen the {S|t} symmetry is broken by the altermagnetic bandsplitting (see Table I). Importantly, although 
o,znk and 
e,znk canbe viewed as two independent contributions, both of them areinduced by the relativistic SO interaction (and the Néel field).Therefore AHE can exist without band splitting. However,it cannot exist without SO interaction, as was discussed inSec. V A 1.It is instructive to briefly discuss the relative importanceof 
o,znk and 
e,znk in some characteristic materials using thesimplified expression 
e,znk /
o,znk ∼ δεmaxn /B anticipated in thestrong coupling limit [28], where εmaxn is a characteristic(maximal) spin splitting near the Fermi level (a measure ofelectron hopping responsible for this splitting). Particularly,an exceptionally large δεmaxn = 0.8 eV has been reported inMnTe on the basis of ARPES measurements [22]. However, Bin MnTe is also expected to be large. In the local spin densityapproximation, it can be estimated as B ∼ 12 JM, where J is acharacteristic exchange coupling (typically being of the orderof intraatomic Hund’s rule coupling, J ∼ 0.8 eV in the caseof Mn2+ [77]) and M is the local spin magnetic moment(M ∼ 4.25μB in MnTe [78]). Therefore the ratio δεmaxn /B canbe estimated as 0.47. The effects of on-site Coulomb repulsionU are also anticipated [78,79]. Particularly, similarly to theone-orbital model, U can contribute to B and further decreasethe ratio δεmaxn /B.RuO2 is an exceptional case. According to theoreticalcalculations [18], εmaxn is large (∼1 eV), while M is small(∼1.17μB). Furthermore, J for 4d materials is also expectedto be relatively small (∼0.6 eV [77]). Then, δεmaxn /B can beestimated as 2.85. This value will be somewhat decreased byconsidering the effects of on-site U , which are also anticipatedin RuO2 [18]. In any case, the even-order contribution 
e,znk isexpected to be large in RuO2. However, it does not necessarilymean that the odd-order contribution 
o,znk is negligible. Fur-thermore, whether 
e,znk will dominate in the calculations ofσxy depends on how the topology of 
e,znk matches that of theFermi surface and whether the areas of large spin splitting ofbands across the Fermi level will coincide with the ones where
e,znk is large [28].VIII. SUMMARY AND OUTLOOKThe time-reversal symmetry breaking in AFM substancesimplies certain analogies with ferromagnetism. On the onehand, it can lift the Kramers degeneracy and split the AFMbands with opposite spins. This is an overt analogy, whichrelies on a more or less standard picture of antiferromagnetismwhere there are two magnetic sublattices with the momentspointing in opposite directions. Today, this analogy is inten-sively explored in the context of altermagnetism [2–4].In the present work, we have argued that there is a deeperanalogy, which allows us to present certain classes of antifer-romagnets with broken T as if they were ferromagnets withonly one magnetic site per cell.Such a representation appears to be possible due to thehidden {S|t} symmetry of SO interaction in centrosymmetricantiferromagnets, allowing transformation to the localcoordinate frame where all the spins are pointed alongthe positive direction of z, i.e., like in ferromagnets. Thissymmetry is the consequence of the inversional invarianceof the system. The key point is that the inversion symmetrycan be broken locally, in individual bonds. Nevertheless, incentrosymmetric antiferromagnets, some inversion centerscontinue to exist, so that I transforms the AFM lattice toitself. If the system is subjected to a lattice distortion, inorder to preserve I, this distortion must be antipolar. In otherwords, we are dealing with antipolar antiferromagnetism.The antipolar distortion inevitably leads to the doubling ofthe crystallographic cell, where the SO interaction behaves asan antiferromagnetically ordered object and changes its signwhen one moves from one magnetic sublattice to another.Nevertheless, this sign change of the SO interaction can becompensated in some local coordinate frame.Indeed, the {S|t} symmetry allows us to use the generalizedBloch theorem, which greatly simplifies the first-principlescalculations of incommensurate spin-spiral states [31,54]. Us-ing the generalized Bloch theorem, any such spiral with anarbitrary propagation vector q can be described within onecrystallographic unit cell. The AFM alignment of spins is aparticular case of the spin spiral. Nevertheless, the generalizedBloch theorem is typically at odds with the relativistic SOinteraction, which tends to lock the spins along one particu-lar direction and, thus, acts against the spin spiral [74]. Theantiferromagnetism appears to be a special case: even thoughthe generalized Bloch theorem is not valid for an arbitrarypropagation q, it can still be valid for the AFM alignment inan antipolar lattice, which imposes a constraint on the form ofthe SO interaction. As a result, in the local coordinate frame,the Néel field and SO interaction become translationally in-variant on the “ferromagnetic” lattice with one magnetic siteper cell.The regular AFM order doubles the unit cell. In unconven-tional antiferromagnets with broken T , the magnetic unit cellshould be as large as the crystallographic one [1]. We substan-tially revise this canonical point of view and show that, due tothe {S|t} symmetry, the magnetic unit cell of centrosymmetricantiferromagnets can be even smaller than the crystallo-graphic one. The situation is highly unusual, but reflects thefundamental symmetry of these systems and naturally ex-plains their similarity with ferromagnets. The emergence ofAHE, orbital magnetization, and magneto-optical effect in thiscase becomes natural, and all these “ferromagnetic” propertiescan be evaluated in the same way as for regular ferromagnets,but in the local coordinate frame.Because the {S|t} symmetry protects the AFM band degen-eracy, this case is qualitatively distinct from altermagnetism.These phenomena are related to different symmetries, whichtypically coexist in one magnetic space group: the {S|t} sym-metry is a consequence of inversional invariance, while theband splitting occurs due to rotational, typically {C|t}, sym-metry, which is combined with T . {C|t} does not conflict withthe {S|t} symmetry of the SO interaction, which operates be-tween different sublattices. The main effect of {C|t} on the SOinteraction is that it relates the parameters of this interactionon the symmetry-related bonds around each magnetic site.However, {C|t} also specifies the form of hopping parametersoperating in different sublattices, which are related by the154413-13I. V. SOLOVYEV PHYSICAL REVIEW B 114, 154413 (2026)rotation C. This can split the AFM bands and eventually breakthe {S|t} symmetry.If the microscopic Hamiltonian were real, the trans-formation {S|t} would be identical to {T |t}. However,when the bonds connecting different sublattices are non-centrosymmetric, as in antipolar lattices, the Hamiltonian iscomplex. Then, {S|t} is not the same as {T |t}. Thereforethe time-reversal symmetry can be broken, but the Hamil-tonian remains invariant under {S|t}, keeping the bandsdegenerate. This is another fundamental reason why the cen-trosymmetric antiferromagnets can exhibit AHE and otherferromagnetic phenomena even if the bands are spin de-generate. The fundamental {S|t} symmetry establishes theconnection between unconventional antiferromagnetism withspin-degenerate bands and conventional ferromagnetism withspin-split bands.The basic idea was illustrated on a number of ex-amples, including a square perovskite lattice, monoclinicVF4 and CuF2, and antiferromagnetism in the rutile com-pounds. Nevertheless, it has much wider implications forthe properties of unconventional antiferromagnets. For in-stance, the authors of a recent review article [25] haveproposed 221 three-dimensional materials, which could po-tentially be altermagnetic, and this number continues to grow.However, most of these materials are centrosymmetric antifer-romagnets, whose fundamental ferromagnetic properties areconsequences of the hidden {S|t} symmetry.The band splitting provides an additional contribution toAHE, which is also related to relativistic SO interaction.However, in model considerations, it emerges only as a smallcorrection to AHE obtained for the spin-degenerate bands[28]. Moreover, for certain materials, such as La2CuO4, theband splitting vanishes due to the special symmetry constraintimposed on parameters of the one-orbital model, which isexpected to provide an adequate description near the Fermilevel [66,67].In light of these arguments, systematic experimental infor-mation establishing a correlation between the band splittingand AHE is highly welcome. Today, this information is verysparse. For instance, AHE has been experimentally mea-sured in MnTe [79] and attributed to the large band splitting[22,23,79]. AHE has been also reported in AFM FeS [80],but no information about the band splitting is currently avail-able. As an opposite example, no indications of magnonband splitting have been reported in MnF2 [81], which isyet another altermagnetic candidate crystallizing in the ru-tile structure [25]. Although this is not a direct probe ofthe band splitting, some relationship is anticipated betweenthem because interatomic exchange interactions should re-flect details of the electronic structure [33]. Our work clearlyshows that MnF2 can exhibit the magneto-optical effect orAHE (for instance, under doping) regardless of whether thereis band splitting. An interesting aspect of MnF2 is that thed5 configuration of Mn2+ is nearly spherical. Therefore thealternation of exchange interactions (and hopping parameters)responsible for the band splitting is expected to be small,as is the single-ion anisotropy. In this sense, the finding ofRef. [81] is quite natural. Although in the ground state themagnetic moments in MnF2 are aligned parallel to z, whichprohibits the magneto-optical activity, they can be rotatedinto the basal plane via the spin-flop transition in a magneticfield [82,83].The spin group theory is a popular tool for identifyingpossible band splitting in altermagnetic materials [2,84]. Nev-ertheless, the symmetry alone does not say whether thissplitting is large or small. It can be accidentally zero (as inMnF2) or identically zero (as in the one-orbital model ofLa2CuO4). On the other hand, the SO interaction, which leadsto AHE in antipolar antiferromagnets, is always finite dueto the lack of bond inversion symmetry. Formally, we dealwith the same materials, which were recently identified asthe novel type-III phase of magnetism (altermagnetism) [2].However, these materials have a long history [5,6] and wereknown before as centrosymmetric antiferromagnets [8,12].Centrosymmetric antiferromagnetism is a more general def-inition, implying a wide range of “ferromagnetic” properties,which are not necessarily related to the spin splitting of bands.The altermagnetism (the band splitting) is just a subclass ofcentrosymmetric antiferromagnetism.The lack of inversion symmetry would break the {S|t}symmetry of the SO interaction. For instance, in the simplestpolar magnets, the SO interaction is translationally invariantand transforms to itself under t. This induces DM interac-tions connecting different unit cells, which are responsiblefor the formation of noncollinear magnetic textures, suchas spin spirals or skyrmions [38–40]. Thus generally suchsystems cannot be classified as “collinear antiferromagnets.”Nevertheless, the lack of inversion symmetry can lead to anumber of interesting properties, including skyrmion-inducedtopological Hall effect [85].Another prominent example of topological magnetism isthe noncoplanar antiferromagnetism realized in triangular,pyrochlore, or kagome lattices [86–89]. The main differencefrom nearly collinear centrosymmetric antiferromagnetsconsidered in the present work is that in noncoplanarmagnets, AHE is induced by scalar spin chirality without SOinteraction.Thus the altermagnetic splitting of bands is certainlyan interesting theoretical discovery leading to a number ofpractically important phenomena such as the spin-current gen-eration in AFM substances [17,20,26]. However, regarding thewhole spectrum of properties expected in centrosymmetric an-tiferromagnets with broken time-reversal symmetry, it wouldnot be right to attribute all of them to the band splitting andtry to consider all of them only from the viewpoint of thissplitting. Even if the band splitting is small, the material is stillexpected to host robust AHE and net orbital magnetization,exhibit magneto-optical rotations and other “ferromagnetic”phenomena, which are related to other, hidden, symmetries ofsuch an AFM state, being the consequence of the inversionalinvariance.ACKNOWLEDGMENTSI am indebted to Sergey Nikolaev and Akihiro Tanaka forcollaboration on earlier stages of this project [28]. MANA issupported by the World Premier International Research CenterInitiative (WPI), MEXT, Japan.The author declares no competing interests.154413-14HIDDEN FERROMAGNETISM OF CENTROSYMMETRIC … PHYSICAL REVIEW B 114, 154413 (2026)DATA AVAILABILITYThe data that support the findings of this article are notpublicly available. The data are available from the author uponreasonable request.APPENDIX A: ELIMINATION OF WEAKLYFERROMAGNETIC COMPONENTSOF THE SPIN-ORBIT COUPLINGTo be specific, let us consider the case where x and zcomponents of the SO coupling between sublattices 1 and 2have the same sign in all neighboring bonds (and are respon-sible for weak spin ferromagnetism), while the y componentis sign-alternating (and responsible for AHE), which is rathercommon for two-sublattice antiferromagnets [28]. The com-plex transfer integrals, including the SO coupling, are givenby tR,R′ 1̂ + itR,R′ · σ̂.Then, let us consider the unitary transformation of sublat-tice 2, Û = eiϕn·σ̂ = cos ϕ2 1̂ + in · σ̂ sin ϕ2 , such that (tR,R′ 1̂ +itR,R′ · σ̂)Û ≈ t̃R,R′ 1̂ + it yR,R′ σ̂y. In principle, by using such atransformation, one can completely eliminate the SO termitR,R′ · σ̂ separately in each bond [32,55]. However, this can-not be done simultaneously for all the bonds unless theparameters of the SO coupling are the same in all these bonds.Nevertheless, one can try to eliminate t xR,R′ ≡ tx and t zR,R′ ≡ tz,which are the same in all the bonds.Assuming n = (tx,0,tz )√t2x +t2z, which does not depend on the alter-nating y-component, one can find the following equations forϕ:tx cosϕ2+ nxt sinϕ2− [tR,R′ × n]x sinϕ2= 0andtz cosϕ2+ nzt sinϕ2− [tR,R′ × n]z sinϕ2= 0,where [tR,R′ × n]x = t yR,R′nz and [tR,R′ × n]z = −t yR,R′nx. Thefirst two terms in these equations are first order in the SOcoupling, whereas the third term is second order and can beneglected (note that ϕ is first order). Therefore one can findthat ϕ = −2 arctan√t2x +t2zt .APPENDIX B: ELECTRONIC STRUCTURE AND MINIMALMODEL FOR VF4 AND CuF2The electronic structure of VF4 and CuF2 in LDA with theexperimental lattice parameters [68,69] is shown in Fig. 9.The calculations are performed using the linear muffin-tinorbital (LMTO) method [90,91] and Vosko-Wilk-Nusairparametrization for the exchange-correlation potential inLDA [92].FIG. 9. Electronic structure of VF4 (left) and CuF2 (right) inLDA. The Fermi level is at zero energy. Shaded areas depict thebands, which were used for the construction of minimal one-orbitalmodels.In both cases, the electronic structure is characterized bytwo well-isolated bands at the Fermi level, which are depictedin Fig. 9. For each spin, these bands can accommodate oneelectron. Therefore, in nonmagnetic LDA, the bands are half-filled.Using Bloch functions for these bands, one can constructthe localized Wannier functions, which serve as the basisfor a minimal model. For these purposes, we employ theprojector-operator technique [75,76]. Then, the transfer in-tegrals tR,R′ 1̂ + itR,R′ · σ̂ can be identified with the matrixelements of the LDA Hamiltonian with the SO coupling inthe Wannier basis. The screened on-site Coulomb repulsionU can be evaluated within the constrained random-phase ap-proximation [93], as explained in Ref. [75]. At half-filling, theNéel field is related to U as B ≈ U/2.The transfer integrals operating within the same sublatticeswere additionally averaged. Namely, using the transfer inte-grals in sublattices I and II, one can define tR,R′ = 12 (t IR,R′ +t IIR,R′ ) (the regular contribution) and δtR,R′ = 12 (t IR,R′ − t IIR,R′ )(the altermagnetic contribution). The approximation consistsin neglecting δtR,R′ . This suppresses the spin splitting of theAFM bands, but allows us to employ the generalized Blochtheorem and describe these AFM systems as if they had onlyone magnetic site in the unit cell. The validity of this ap-proximation depends on the system: while δtR,R′ and tR,R′ arecomparable in CuF2, δtR,R′ appears to be much smaller thantR,R′ in VF4 (see Fig. 10).FIG. 10. Distance dependence of regular (tR,R′ ) and altermag-netic (δtR,R′ ) transfer integrals operating within the same sublatticesin VF4 and CuF2.154413-15I. V. SOLOVYEV PHYSICAL REVIEW B 114, 154413 (2026)[1] I. E. Dzyaloshinskii, Space and time parity violation in anyonicand chiral systems, Phys. Lett. A 155, 62 (1991).[2] L. Šmejkal, J. Sinova, and T. Jungwirth, Beyond conventionalferromagnetism and antiferromagnetism: A phase with nonrel-ativistic spin and crystal rotation symmetry, Phys. Rev. X 12,031042 (2022).[3] L. Šmejkal, J. Sinova, and T. Jungwirth, Emerging researchlandscape of altermagnetism, Phys. Rev. X 12, 040501 (2022).[4] I. Mazin, Editorial: Altermagnetism—A new punch line of fun-damental magnetism, Phys. Rev. X 12, 040002 (2022).[5] I. Dzyaloshinsky, A thermodynamic theory of “weak” ferro-magnetism of antiferromagnetics, J. Phys. Chem. Solids 4, 241(1958).[6] I. E. Dzyaloshinskii, The problem of piezomagnetism, Zh Eksp.Teor. Fiz. 33, 807 (1957) [JETP (USSR) 6, 621 (1958)].[7] I. E. Dzyaloshinskii, On the magneto-electrical effect in atifer-romagnets, Zh Eksp. Teor. Fiz. 37, 881 (1959) [JETP (USSR)10, 628 (1960)].[8] E. A. Turov, Kinetic, Optical, and Acoustic Properties of An-tiferromagnets (Ural Division of Academy of Sciences of theUSSR, Sverdlovsk, 1990).[9] E. A. Turov and V. G. Shavrov, On some galvano- and thermo-magnetic effects in antiferromagnets, Zh Eksp. Teor. Fiz. 43,2273 (1962) [JETP (USSR) 16, 1606 (1963)].[10] P. A. McClarty and J. G. Rau, Landau theory of altermagnetism,Phys. Rev. Lett. 132, 176702 (2024).[11] L. Šmejkal, A. H. MacDonald, J. Sinova, S. Nakatsuji, and T.Jungwirth, Anomalous Hall antiferromagnets, Nat. Rev. Matter.7, 482 (2022).[12] E. A. Turov, Can the magnetoelectric effect coexist with weakpiezomagnetism and ferromagnetism? Usp. Fiz. Nauk 164, 325(1994); Can the magnetoelectric effect coexist with weak piezo-magnetism and ferromagnetism? Phys.-Usp. 37, 303 (1994).[13] T. Yamaguchi and K. Tsushima, Magnetic symmetry of rare-earth orthochromites and orthoferrites, Phys. Rev. B 8, 5187(1973).[14] Y. Tokunaga, N. Furukawa, H. Sakai, Y. Taguchi, T. Arima, andY. Tokura, Composite domain walls in a multiferroic perovskiteferrite, Nat. Mater. 8, 558 (2009).[15] T. Okugawa, K. Ohno, Y. Noda, and S. Nakamura, Weaklyspin-dependent band structures of antiferromagnetic perovskiteLaMO3 (M = Cr, Mn, Fe), J. Phys.: Condens. Matter 30,075502 (2018).[16] S. Hayami, Y. Yanagi, and H. Kusunose, Momentum-dependentspin splitting by collinear antiferromagnetic ordering, J. Phys.Soc. Jpn. 88, 123702 (2019).[17] M. Naka, S. Hayami, H. Kusunose, Y. Yanagi, Y. Motome, andH. Seo, Spin current generation in organic antiferromagnets,Nat. Commun. 10, 4305 (2019).[18] L. Šmejkal, R. González-Hernández, T. Jungwirth, and J.Sinova, Crystal time-reversal symmetry breaking and sponta-neous Hall effect in collinear antiferromagnets, Sci. Adv. 6,eaaz8809 (2020).[19] M. Naka, S. Hayami, H. Kusunose, Y. Yanagi, Y. Motome, andH. Seo, Anomalous Hall effect in κ-type organic antiferromag-nets, Phys. Rev. B 102, 075112 (2020).[20] H.-Y. Ma, M. Hu, N. Li, J. Liu, W. Yao, J.-F. Jia, and J. Liu,Multifunctional antiferromagnetic materials with giant piezo-magnetism and noncollinear spin current, Nat. Commun. 12,2846 (2021).[21] M. Naka, Y. Motome, and H. Seo, Anomalous Hall effectin antiferromagnetic perovskites, Phys. Rev. B 106, 195149(2022).[22] T. Osumi, S. Souma, T. Aoyama, K. Yamauchi, A. Honma, K.Nakayama, T. Takahashi, K. Ohgushi, and T. Sato, Observationof a giant band splitting in altermagnetic MnTe, Phys. Rev. B109, 115102 (2024).[23] K. D. Belashchenko, Giant strain-induced spin splitting effect inMnTe, a g-wave altermagnetic semiconductor, Phys. Rev. Lett.134, 086701 (2025).[24] M. Hu, X. Cheng, Z. Huang, and J. Liu, Catalog of C-pairedspin-momentum locking in antiferromagnetic systems, Phys.Rev. X 15, 021083 (2025).[25] L. Bai, W. Feng, S. Liu, L. Šmejkal, Y. Mokrousov, and Y.Yao, Altermagnetism: Exploring new frontiers in magnetismand spintronics, Adv. Funct. Mater. 34, 2409327 (2024).[26] M. Naka, Y. Motome, and H. Seo, Altermagnetic perovskites,npj Spintron. 3, 1 (2025).[27] T. Moriya, Anisotropic superexchange interaction and weakferromagnetism, Phys. Rev. 120, 91 (1960).[28] I. Solovyev, S. Nikolaev, and A. Tanaka, Altermagnetism andweak ferromagnetism, npj Quantum Mater. 11, 54 (2026).[29] H. Chen, Q. Niu, and A. H. MacDonald, Anomalous Hall effectarising from noncollinear antiferromagnetism, Phys. Rev. Lett.112, 017205 (2014).[30] Strictly speaking, we consider a specific situation when themagnetic sites are located in the inversion centers. Anotherpossibility is when the magnetic site itself is not in the cen-trosymmetric position, but I connects two magnetic sites [8].Such a situation is realized, for instance, in Fe2O3 [5]. Never-theless, we note that in the latter case, one can treat the magneticmolecule, consisting of two Fe atoms across the inversion cen-ter, as a formal lattice site. Then, the situation will be reducedto the one considered in the present work.[31] L. M. Sandratskii, Noncollinear magnetism in itinerant-electronsystems: Theory and applications, Adv. Phys. 47, 91 (1998).[32] L. Shekhtman, O. Entin-Wohlman, and A. Aharony, Moriya’sanisotropic superexchange interaction, frustration, andDzyaloshinsky’s weak ferromagnetism, Phys. Rev. Lett. 69,836 (1992).[33] I. V. Solovyev, Linear response theories for interatomic ex-change interactions, J. Phys.: Condens. Matter 36, 223001(2024).[34] I. V. Solovyev, Magneto-optical effect in the weak ferromagnetsLaMO3 (M = Cr, Mn, and Fe), Phys. Rev. B 55, 8060 (1997).[35] H. Katsura, N. Nagaosa, and A. V. Balatsky, Spin current andmagnetoelectric effect in noncollinear magnets, Phys. Rev. Lett.95, 057205 (2005).[36] I. Solovyev, R. Ono, and S. Nikolaev, Magnetically inducedpolarization in centrosymmetric bonds, Phys. Rev. Lett. 127,187601 (2021).[37] I. V. Solovyev, Basic aspects of ferroelectricity induced bynoncollinear alignment of spins, Condens. Matter 10, 21(2025).[38] I. E. Dzyaloshinskii, Theory of helicoidal structures in antifer-romagnets. I. Nonmetals, Zh Eksp. Teor. Fiz. 46, 1420 (1964)[JETP (USSR) 19, 960 (1964)].[39] A. N. Bogdanov and D. A. Yablonskii, Thermodynamicallystable “vortices” in magnetically ordered crystals. The mixedstate of magnets, Sov. Phys. JETP 68, 101 (1989).154413-16https://doi.org/10.1016/0375-9601(91)90508-6https://doi.org/10.1103/PhysRevX.12.031042https://doi.org/10.1103/PhysRevX.12.040501https://doi.org/10.1103/PhysRevX.12.040002https://doi.org/10.1016/0022-3697(58)90076-3https://jetp.ras.ru/cgi-bin/dn/e_006_03_0621.pdfhttps://jetp.ras.ru/cgi-bin/dn/e_010_03_0628.pdfhttps://jetp.ras.ru/cgi-bin/dn/e_016_06_1606.pdfhttps://doi.org/10.1103/PhysRevLett.132.176702https://doi.org/10.1038/s41578-022-00430-3https://doi.org/10.3367/UFNr.0164.199403f.0325https://doi.org/10.1070/PU1994v037n03ABEH000015https://doi.org/10.1103/PhysRevB.8.5187https://doi.org/10.1038/nmat2469https://doi.org/10.1088/1361-648X/aa9e70https://doi.org/10.7566/JPSJ.88.123702https://doi.org/10.1038/s41467-019-12229-yhttps://doi.org/10.1126/sciadv.aaz8809https://doi.org/10.1103/PhysRevB.102.075112https://doi.org/10.1038/s41467-021-23127-7https://doi.org/10.1103/PhysRevB.106.195149https://doi.org/10.1103/PhysRevB.109.115102https://doi.org/10.1103/PhysRevLett.134.086701https://doi.org/10.1103/PhysRevX.15.021083https://doi.org/10.1002/adfm.202409327https://doi.org/10.1038/s44306-024-00066-9https://doi.org/10.1103/PhysRev.120.91https://doi.org/10.1038/s41535-026-00900-9https://doi.org/10.1103/PhysRevLett.112.017205https://doi.org/10.1080/000187398243573https://doi.org/10.1103/PhysRevLett.69.836https://doi.org/10.1088/1361-648X/ad215ahttps://doi.org/10.1103/PhysRevB.55.8060https://doi.org/10.1103/PhysRevLett.95.057205https://doi.org/10.1103/PhysRevLett.127.187601https://doi.org/10.3390/condmat10020021https://jetp.ras.ru/cgi-bin/dn/e_019_04_0960.pdfhttps://jetp.ras.ru/cgi-bin/dn/e_068_01_0101.pdfHIDDEN FERROMAGNETISM OF CENTROSYMMETRIC … PHYSICAL REVIEW B 114, 154413 (2026)[40] U. K. Rößler, A. N. Bogdanov, and C. Pfleiderer, Spontaneousskyrmion ground states in magnetic metals, Nature (London)442, 797 (2006).[41] I. V. Solovyev, Magnetic structure of the noncentrosymmetricperovskites PbVO3 and BiCoO3: Theoretical analysis, Phys.Rev. B 85, 054420 (2012).[42] G. Catalan and J. F. Scott, Physics and applications of bismuthferrite, Adv. Mater. 21, 2463 (2009).[43] I. Kézsmárki, S. Bordács, P. Milde, E. Neuber, L. M. Eng,J. S. White, H. M. Rønnow, C. D. Dewhurst, M. Mochizuki, K.Yanai, H. Nakamura, D. Ehlers, V. Tsurkan, and A. Loidl, Néel-type skyrmion lattice with confined orientation in the polarmagnetic semiconductor GaV4S8, Nat. Mater. 14, 1116 (2015).[44] S. A. Nikolaev and I. V. Solovyev, Microscopic theory of elec-tric polarization induced by skyrmionic order in GaV4S8, Phys.Rev. B 99, 100401(R) (2019).[45] H. T. Yi, Y. J. Choi, S. Lee, and S.-W. Cheong, Multiferroicity inthe square-lattice antiferromagnet of Ba2CoGe2O7, Appl. Phys.Lett. 92, 212904 (2008).[46] H. Murakawa, Y. Onose, S. Miyahara, N. Furukawa, andY. Tokura, Ferroelectricity induced by spin-dependent metal-ligand hybridization in Ba2CoGe2O7, Phys. Rev. Lett. 105,137202 (2010).[47] T. Nakajima, Y. Tokunaga, V. Kocsis, Y. Taguchi, Y. Tokura, andT.-h. Arima, Uniaxial-stress control of spin-driven ferroelectric-ity in multiferroic Ba2CoGe2O7, Phys. Rev. Lett. 114, 067201(2015).[48] I. V. Solovyev, Magnetization-induced local electric dipolesand multiferroic properties of Ba2CoGe2O7, Phys. Rev. B 91,224423 (2015).[49] A. Zheludev, G. Shirane, Y. Sasago, N. Koide, and K.Uchinokura, Spiral phase and spin waves in the quasi-two-dimensional antiferromagnet Ba2CuGe2O7, Phys. Rev. B 54,15163 (1996).[50] H. Murakawa, Y. Onose, and Y. Tokura, Electric-field switchingof a magnetic propagation vector in a helimagnet, Phys. Rev.Lett. 103, 147201 (2009).[51] R. Ono, S. Nikolaev, and I. Solovyev, Fingerprints of spin-current physics on magnetoelectric response in the spin- 12magnet Ba2CuGe2O7, Phys. Rev. B 102, 064422 (2020).[52] A. N. Bogdanov, U. K. Rößler, M. Wolf, and K.-H. Müller,Magnetic structures and reorientation transitions in noncen-trosymmetric uniaxial antiferromagnets, Phys. Rev. B 66,214410 (2002).[53] T. Thio, T. R. Thurston, N. W. Preyer, P. J. Picone, M. A.Kastner, H. P. Jenssen, D. R. Gabbe, C. Y. Chen, R. J.Birgeneau, and A. Aharony, Antisymmetric exchange and its in-fluence on the magnetic structure and conductivity of La2CuO4,Phys. Rev. B 38, 905 (1988).[54] O. N. Mryasov, A. I. Liechtenstein, L. M. Sandratskii, and V. A.Gubanov, Magnetic structure of FCC iron, J. Phys.: Condens.Matter 3, 7683 (1991).[55] T. A. Kaplan, Single-band Hubbard model with spin-orbit cou-pling, Z. Phys. B 49, 313 (1983).[56] M. Roig, A. Kreisel, Y. Yu, B. M. Andersen, and D. F.Agterberg, Minimal models for altermagnetism, Phys. Rev. B110, 144412 (2024).[57] To be consistent with the current definition, the sign of the SOcoupling in Eq. (16) was changed in comparison with Ref. [28].[58] S. Bhowal and N. A. Spaldin, Ferroically ordered magneticoctupoles in d-wave altermagnets, Phys. Rev. X 14, 011019(2024).[59] T. Sato and S. Hayami, Quantum theory of magnetic octupolein periodic crystals and application to d-wave altermagnets, npjQuantum Mater. 11, 32 (2026).[60] J. Ōiké, R. Peters, and K. Shinada, Thermodynamic formulationof the spin magnetic octupole moment in bulk crystals, Phys.Rev. B 112, 134412 (2025).[61] X. Zhou, W. Feng, R.-W. Zhang, L. Šmejkal, J. Sinova, Y.Mokrousov, and Y. Yao, Crystal thermal transport in altermag-netic RuO2, Phys. Rev. Lett. 132, 056701 (2024).[62] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P.Ong, Anomalous Hall effect, Rev. Mod. Phys. 82, 1539 (2010).[63] Z. Liu, M. Wei, W. Peng, D. Hou, Y. Gao, and Q. Niu, Mul-tipolar anisotropy in anomalous Hall effect from spin-groupsymmetry breaking, Phys. Rev. X 15, 031006 (2025).[64] T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Or-bital magnetization in periodic insulators, Phys. Rev. Lett. 95,137205 (2005).[65] J. Shi, G. Vignale, D. Xiao, and Q. Niu, Quantum theory oforbital magnetization and its generalization to interacting sys-tems, Phys. Rev. Lett. 99, 197202 (2007).[66] M. Hirayama, Y. Yamaji, T. Misawa, and M. Imada, Ab initioeffective Hamiltonians for cuprate superconductors, Phys. Rev.B 98, 134501 (2018).[67] M. T. Schmid, J.-B. Morée, R. Kaneko, Y. Yamaji, andM. Imada, Superconductivity studied by solving ab initiolow-energy effective Hamiltonians for carrier doped CaCuO2,Bi2Sr2CuO6, Bi2Sr2CaCu2O8, and HgBa2CuO4, Phys. Rev. X13, 041036 (2023).[68] S. Becker and B. G. Müller, Vanadiumtetrafluorid, Angew.Chem. 102, 426 (1990).[69] P. C. Burns and F. C. Hawthorne, Rietveld refinement of thecrystal structure of CuF2, Powder Diffr. 6, 156 (1991).[70] T. Kimura, Y. Sekio, H. Nakamura, T. Siegrist, and A. P.Ramirez, Cupric oxide as an induced-multiferroic with high-TC ,Nat. Mater. 7, 291 (2008).[71] N. Terada, D. D. Khalyavin, P. Manuel, F. Orlandi, C. J. Ridley,C. L. Bull, R. Ono, I. Solovyev, T. Naka, D. Prabhakaran,and A. T. Boothroyd, Room-temperature type-II multiferroicphase induced by pressure in cupric oxide, Phys. Rev. Lett. 129,217601 (2022).[72] I. E. Dzyaloshinskii, The magnetic structure of fluorides of thetransition metals, Zh Eksp. Teor. Fiz. 33, 1454 (1957) [JETP(USSR) 6, 1120 (1958)].[73] T. Moriya, Theory of magnetism of NiF2, Phys. Rev. 117, 635(1960).[74] L. M. Sandratskii, Insight into the Dzyaloshinskii-Moriyainteraction through first-principles study of chiral magneticstructures, Phys. Rev. B 96, 024450 (2017).[75] I. V. Solovyev, Combining DFT and many-body methods tounderstand correlated materials, J. Phys.: Condens. Matter 20,293201 (2008).[76] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D.Vanderbilt, Maximally localized Wannier functions: Theory andapplications, Rev. Mod. Phys. 84, 1419 (2012).[77] I. V. Solovyev, P. H. Dederichs, and V. I. Anisimov, Cor-rected atomic limit in the local-density approximation and the154413-17https://doi.org/10.1038/nature05056https://doi.org/10.1103/PhysRevB.85.054420https://doi.org/10.1002/adma.200802849https://doi.org/10.1038/nmat4402https://doi.org/10.1103/PhysRevB.99.100401https://doi.org/10.1063/1.2937110https://doi.org/10.1103/PhysRevLett.105.137202https://doi.org/10.1103/PhysRevLett.114.067201https://doi.org/10.1103/PhysRevB.91.224423https://doi.org/10.1103/PhysRevB.54.15163https://doi.org/10.1103/PhysRevLett.103.147201https://doi.org/10.1103/PhysRevB.102.064422https://doi.org/10.1103/PhysRevB.66.214410https://doi.org/10.1103/PhysRevB.38.905https://doi.org/10.1088/0953-8984/3/39/013https://doi.org/10.1007/BF01301591https://doi.org/10.1103/PhysRevB.110.144412https://doi.org/10.1103/PhysRevX.14.011019https://doi.org/10.1038/s41535-026-00865-9https://doi.org/10.1103/frq1-9xx7https://doi.org/10.1103/PhysRevLett.132.056701https://doi.org/10.1103/RevModPhys.82.1539https://doi.org/10.1103/PhysRevX.15.031006https://doi.org/10.1103/PhysRevLett.95.137205https://doi.org/10.1103/PhysRevLett.99.197202https://doi.org/10.1103/PhysRevB.98.134501https://doi.org/10.1103/PhysRevX.13.041036https://doi.org/10.1002/ange.19901020422https://doi.org/10.1017/S0885715600017310https://doi.org/10.1038/nmat2125https://doi.org/10.1103/PhysRevLett.129.217601https://jetp.ras.ru/cgi-bin/dn/e_006_06_1120.pdfhttps://doi.org/10.1103/PhysRev.117.635https://doi.org/10.1103/PhysRevB.96.024450https://doi.org/10.1088/0953-8984/20/29/293201https://doi.org/10.1103/RevModPhys.84.1419I. V. SOLOVYEV PHYSICAL REVIEW B 114, 154413 (2026)electronic structure of d impurities in Rb, Phys. Rev. B 50,16861 (1994).[78] D. Kriegner, H. Reichlova, J. Grenzer, W. Schmidt, E.Ressouche, J. Godinho, T. Wagner, S. Y. Martin, A. B. Shick,V. V. Volobuev, G. Springholz, V. Holý, J. Wunderlich, T.Jungwirth, and K. Výborný, Magnetic anisotropy in antiferro-magnetic hexagonal MnTe, Phys. Rev. B 96, 214418 (2017).[79] R. D. G. Betancourt, J. Zubáč, R. Gonzalez-Hernandez, K.Geishendorf, Z. Šobáň, G. Springholz, K. Olejník, L. Šmejkal,J. Sinova, T. Jungwirth, S. T. B. Goennenwein, A. Thomas, H.Reichlová, J. Železný, and D. Kriegner, Spontaneous anoma-lous Hall effect arising from an unconventional compensatedmagnetic phase in a semiconductor, Phys. Rev. Lett. 130,036702 (2023).[80] R. Takagi, R. Hirakida, Y. Settai, R. Oiwa, H. Takagi, A.Kitaori, K. Yamauchi, H. Inoue, J. Yamaura, D. Nishio-Hamane, S. Itoh, S. Aji, H. Saito, T. Nakajima, T. Nomoto, R.Arita, and S. Seki, Spontaneous Hall effect induced by collinearantiferromagnetic order at room temperature, Nat. Mater. 24, 63(2025).[81] V. C. Morano, Z. Maesen, S. E. Nikitin, J. Lass, D. G. Mazzone,and O. Zaharko, Absence of altermagnetic magnon band split-ting in MnF2, Phys. Rev. Lett. 134, 226702 (2025).[82] G. P. Felcher and R. Kleb, Antiferromagnetic domains and thespin-flop transition of MnF2, Europhys. Lett. 36, 455 (1996).[83] I. Solovyev, Altermagnetism in MnF2: Band splitting and itsphysical consequences, arXiv:2606.01515.[84] X. Chen, J. Ren, Y. Zhu, Y. Yu, A. Zhang, P. Liu, J. Li, Y. Liu, C.Li, and Q. Liu, Enumeration and representation theory of spinspace groups, Phys. Rev. X 14, 031038 (2024).[85] N. Nagaosa and Y. Tokura, Topological properties and dy-namics of magnetic skyrmions, Nat. Nanotechnol. 8, 899(2013).[86] K. Ohgushi, S. Murakami, and N. Nagaosa, Spin anisotropyand quantum Hall effect in the kagome lattice: Chiral spin statebased on a ferromagnet, Phys. Rev. B 62, R6065 (2000).[87] S. Nakatsuji, N. Kiyohara, and T. Higo, Large anomalous Halleffect in a non-collinear antiferromagnet at room temperature,Nature (London) 527, 212 (2015).[88] W. Feng, J.-P. Hanke, X. Zhou, G.-Y. Guo, S. Blügel, Y.Mokrousov, and Y. Yao, Topological magneto-optical effectsand their quantization in noncoplanar antiferromagnets, Nat.Commun. 11, 118 (2020).[89] H. Watanabe, K. Shinohara, T. Nomoto, A. Togo, and R.Arita, Symmetry analysis with spin crystallographic groups:Disentangling effects free of spin-orbit coupling in emergentelectromagnetism, Phys. Rev. B 109, 094438 (2024).[90] O. K. Andersen, Linear methods in band theory, Phys. Rev. B12, 3060 (1975).[91] O. Gunnarsson, O. Jepsen, and O. K. Andersen, Self-consistentimpurity calculations in the atomic-spheres approximation,Phys. Rev. B 27, 7144 (1983).[92] S. H. Vosko, L. Wilk, and M. Nusair, Accurate spin-dependentelectron liquid correlation energies for local spin densitycalculations: A critical analysis, Can. J. Phys. 58, 1200(1980).[93] F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S.Biermann, and A. I. Lichtenstein, Frequency-dependent localinteractions and low-energy effective models from electronicstructure calculations, Phys. Rev. B 70, 195104 (2004).154413-18https://doi.org/10.1103/PhysRevB.50.16861https://doi.org/10.1103/PhysRevB.96.214418https://doi.org/10.1103/PhysRevLett.130.036702https://doi.org/10.1038/s41563-024-02058-whttps://doi.org/10.1103/PhysRevLett.134.226702https://doi.org/10.1209/epl/i1996-00251-7https://arxiv.org/abs/2606.01515https://doi.org/10.1103/PhysRevX.14.031038https://doi.org/10.1038/nnano.2013.243https://doi.org/10.1103/PhysRevB.62.R6065https://doi.org/10.1038/nature15723https://doi.org/10.1038/s41467-019-13968-8https://doi.org/10.1103/PhysRevB.109.094438https://doi.org/10.1103/PhysRevB.12.3060https://doi.org/10.1103/PhysRevB.27.7144https://doi.org/10.1139/p80-159https://doi.org/10.1103/PhysRevB.70.195104